Name: Letter: PHYS 2419 - University of Virginiapeople.virginia.edu/~ecd3m/2419/Fall2011/manual/manual.pdfPHYSICS 2419 WORKSHOP MANUAL Contents Page Introduction 1 Lab 1 Electrostatics

Name: Letter:

PHYS 2419

FALL 2011

COURSE WEBSITE:

http://people.virginia.edu/~ecd3m/2419/Fall2011/

LABORATORY MANUAL

DEPARTMENT OF PHYSICS

UNIVERSITY OF VIRGINIA

COURSE INSTRUCTOR:

E. CRAIG DUKES

EMAIL: [email protected]

LABORATORY SUPERVISOR:

LARRY SUDDARTH


PHYSICS 2419 WORKSHOP MANUAL

Contents Page

Introduction ..................................................................................................................... 1

Lab 1 Electrostatics .............................................................................................. L01-1

Lab 2 Gauss’ Law ................................................................................................ L02-1

Lab 3 Simple DC Circuits . ................................................................................... L03-1

Lab 4 Ohm’s Law & Kirchhoff’s Circuit Rules ................................................... L04-1

Lab 5 Capacitors & RC Circuits . .......................................................................... L05-1

Lab 6 Electron Charge-to-Mass Ratio ................................................................... L06-1

Lab 7 Inductors & LR Circuits ............................................................................. L07-1

Lab 8 AC Currents & Voltages ............................................................................ L08-1

Lab 9 AC Filters & Resonance ............................................................................. L09-1

Lab 10 Geometrical Optics . .................................................................................... L10-1

Lab 11 Polarization ................................................................................................ L11-1

Lab 12 Interference . ............................................................................................... L12-1

Appendix A: Selected Constants ................................................................................. A-1

Appendix B: Graphical Analysis ................................................................................. B-1

Appendix C: Accuracy of Measurements and Treatment of Experimental Uncertainty . C-1

Appendix D: Lasers .................................................................................................... D-1

1

University of Virginia Physics Department PHYS 2419, Fall 2011

PHYSICS 2419 WORKSHOP

INTRODUCTION

FACULTY COORDINATOR: CRAIG DUKES


WEBSITE: http://people.virginia.edu/~ecd3m/2419/Fall2011/home.html

WORKSHOP GOALS AND PHILOSOPHY

Physics is an experimental science. Experiments are performed to test the predictions of

theories or to present data the theories cannot explain in order to spur better theories.

If you find physics difficult, you are not alone. The concepts are often not easy to grasp.

We must each construct our own models of understanding. Passive listening to lectures

and rote memorization are not good ways to learn. We must be able to assimilate the

concepts and apply them to predict further phenomena. Studies have shown that learning

improves when a student thinks about a concept or problem by him/herself first and then

discusses it with a small group of peers. That is the philosophy we will follow in this

workshop. The abilities to work within a group of peers and to communicate ideas, both

orally and in writing, are important skills to have. These are fundamental goals of this

workshop.

Most of the experiments in this workshop will utilize data sensors interfaced to a

computer. We utilize PASCO’s Data Studio software, because of its powerful ability to

take, present, and analyze data. You will find most of the analysis tools you need in Data

Studio. You can find the area, highlight a particular region, find averages, or a host of

other things with Data Studio. You will find that you will normally be able to fit or

model data with an analytic function. We will also make frequent use of Excel.

PURPOSE OF THE COURSE

The purpose of this workshop is to

1. teach you some important physical phenomena and concepts,

2. introduce you to proper laboratory procedures, to use computers and data sensors,

and teach you some basic laboratory techniques,

3. give you confidence in your ability to take measurements and adequately analyze and interpret data,

4. teach you better oral and written communication skills,

5. teach you to think for yourself and to work in groups of peers.

REGISTRATION

Physics 2419 is a dependent course for Physics 2415, but it is not part of Physics 2415. It

is a one-credit course with an independent grade. You must, however, be registered in a

2415 (lecture) section before SIS will allow you to register for a 2419 (workshop)

section.

2 Introduction


If you already have credit for 2415, you must still register for 2415 before SIS will allow

you to sign up for a 2419 section. After you have registered for the lab, you can drop the

lecture.

Registration in Physics 2419 will be blocked on Friday, August 26, 2011. The labs

start on Monday during the semester’s first full week of classes (August 29, 2011).

Once registration is blocked, Mr. Larry Suddarth (room 214 – Physics building, 924-

6843, [email protected]) will be the only person who can add you into a section of

2419.

In that first week, you must attend the section of your choice on time. If you are

registered for that section, your place in that section is secure. If you do not attend or are

late to your registered section, your name will be dropped from that section’s enrollment.

Let us re-emphasize this point: If you are registered for a section and wish to secure

your place in that section, you must attend that section on time during the first full

week of classes.

After students registered for a section (who show up on time!) have been added to the

roster, those who wish to add to that section will then be added if space is available.

Preference will be given to waitlisted students. Since only 24 students may be in any

given section, if more students want to add than there is space available, names will be

drawn at random and added to the roster until the 24 spaces are filled. The remaining

students must find other sections to attend. Note, however, that there are normally two

sections being held simultaneously so that most time slots have space available for 48

students in the two sections.

In the extraordinary event that you cannot attend any sections during that first full week

of classes (say due to major illness or a family emergency), please contact Mr. Suddarth

as soon as possible, but absolutely before your scheduled section. Contact Mr. Suddarth

regarding any problems with registration.

It is your responsibility to be registered for a workshop. If you are unable to find a

workshop open that meets your schedule, go to a suitable section the first week to see if

space becomes available or to see if someone will switch with you. You may need to go

to several workshops before this is successful.

COURSE ORGANIZATION

Your work in Physics 2419 will consist of three parts:

1. A pre-lab homework that you must complete before coming to the lab.

2. The lab itself, answering all the questions and predictions, and attaching data,

results, graphs, and analysis as requested with your group members that will be

turned in at the end of the lab.

3. A post-lab quiz that you must finish in the specified time period.

The pre-lab homework and post-lab quiz will be done on the WebAssign Internet site.

The labs meet during each full week of classes and are overseen by a graduate teaching

assistant (commonly called a TA). The TA’s responsibilities are to ensure the safety of

Introduction 3


the students, protect the equipment, provide good teaching pedagogy to help you learn as

much as possible, provide additional instructions and information concerning the lab,

grade your work and, together with the faculty, assign your grade.

CLASS WEBSITE

The course website

http://people.virginia.edu/~ecd3m/2419/Fall2011/home.html

has up-to-date information related to the organization of the course. Some of that

information is summarized below. Please consult the class website for TA contact

information, a list of the lab sessions, their rooms and instructors, office hours, etc.

LABORATORY MANUAL

Every student must purchase the manual for Physics 2419 at the UVa bookstore.

This manual contains the workshop activities which you will use each week. You will be

assessed a 10% penalty if you fail to bring your manual to lab.

GRADING POLICY

The workshop will be graded as follows:

• The pre-lab homework is worth 20%.

• The weekly lab is worth 40%. Your grade is based on your performance in the

laboratory as evidenced by what you turn in each week.

• The post-lab quiz is worth 40%.

No scores will be dropped. Lab scores will be curved based on your TA’s students only

(to take account of the different TA grading scales). Final grades are determined by

relative class “rank”, not by a predefined numerical scale. Historically, the average grade

in 2419 has been between B and B+.

PREPARATION

Before attending your lab section during the first full week of classes, look over the lab

manual and become familiar with the appendices to which you should refer as needed

throughout the semester. Particularly important is Appendix D: The Accuracy of

Measurements and Significant Figures. Refer to Appendix D and apply it appropriately

throughout the semester.

For each lab, you must do the pre-lab homework that can be found on the WebAssign

Internet site:

https://www.webassign.net/uva/login.html

We are not having you submit a formal written lab report, but instead, we are requiring

you to spend time preparing for the lab each week. We expect that since you are better

prepared, the lab will be a better learning experience.

In order to prepare for the lab each week, do the following:

4 Introduction


1. Read over the lab write-up in this manual (including the relevant appendices) to

get an overview of the material.

2. Read the instructions again, but this time more carefully; highlighting the

important features of the lab. Try to work through any derivations you do not

understand (refer to your textbook as needed). In other words, be an active reader

and study the manual.

3. Complete the pre-lab homework. The homework is not pledged and you are

encouraged to work together to understand and solve the problems. However,

you are responsible for really knowing how to work the problems. Simply

“plugging numbers” into a formula or spreadsheet given to you will teach you

nothing.

PROCEDURE IN THE LAB

Normally you will work in groups of three. You will be assigned to a different group

each week. We encourage a free exchange of ideas between group members (and also

generally in the laboratory), and we expect you to share both in taking data and in

operating the computer system. You will turn in your lab materials as a group at the end

of the period. Everyone must fill out the material asked for in the manual, but only turn

in one set of graphs and data when you are asked to print them out. Be sure that all such

printouts are well noted with the activity number and your lab partners’ names. You and

your group members will not necessarily receive a common grade for the lab each week,

because we will grade both your results and your answers. Each lab is two hours

(technically one hour and fifty minutes) long. You are expected to have vacated the room

within one hour and fifty-five minutes to allow the next section to begin on time.

ABSENCES AND TARDINESS

Absences will be excused only for legitimate reasons (illness, a death in your family,

etc.). If you must miss a laboratory session, submit a written petition (email will suffice)

to your TA explaining your situation and requesting permission to make up the lab. This

request should be made within 48 hours of your scheduled lab period. Unexcused

absences earn a grade of zero for that lab.

An exam (or study session) for another course is NOT an approved reason to miss lab. If

one of your other professors schedules an exam for the time that you have lab, you should

inform them that you already have a class scheduled for that time and ask them to make

appropriate arrangements. [You should, of course, do this as soon as the exam schedule

is communicated to you.]

Late arrival for any lab session is very disruptive and will be penalized. After an initial

five minute grace period, the TA will deduct 10% from your grade for the first ten

minutes of tardiness and 15% for each successive 10-minute period (or part thereof).

LAB MAKE-UPS

You must receive written permission from your TA to make up a missed lab. All make-

ups must be arranged by your TA in advance. Without prior arrangements, there will be

Introduction 5


a 50% penalty (assuming, of course, that there is even space available in the make-up

session). Make-up labs are normally Thursday afternoon in Physics room 215 beginning

at 4 PM. You may only make up a lab during the week that you missed it or the

following week. The labs are not left set up more than the following week. It is the

student’s responsibility to make sure the TA has given permission and to attend the

make-up.

If you miss a lab for an approved reason, and do not make up the lab until the following

week, your TA should contact Mr. Suddarth for an extension for the homework and the

quiz. If you take the make-up on Thursday of the regular lab week, you do not receive a

time extension.

WEBASSIGN POLICY

Please pay close attention to the due dates of the WebAssign pre- and post-lab

assignments.

The pre-lab homework will typically be posted on the Tuesday of the week before the

regularly scheduled lab. The homework is due thirty minutes before the lab (and no

extension other than that described in the make-up policy will be granted).

You will be given several submissions to obtain the correct answer. Do not waste your

submissions. Seek assistance if you are having difficulty. Remember, the homework is

NOT pledged. Indeed, you are encouraged to work together. As noted earlier, though,

you are expected to learn how to do the problem, not just “work the calculator”.

The post-lab quiz IS pledged. You are allowed to use your book, notes, and manual

(available in PDF format via the class web site), but you are NOT allowed to consult

anyone.

IMPORTANT POINT: The quiz will be posted right after the lab (at the next hour

mark). It is DUE by midnight of that same day. We do, however, give you a penalty-

free extension until the following “calendar” Monday at 5 AM. This is an absolute

deadline: If, for whatever reason, you do not complete the post-lab quiz on time, you

will receive a zero.

ANOTHER IMPORTANT POINT: The quizzes have a time limit of forty-five

minutes. To allow for “transit delays” and the like, we will accept submissions up to five

minutes late (again, without penalty). Like the 5 AM Monday deadline, this is an

absolute deadline and if you if you do not submit the quiz on time, you will receive a

zero. Do NOT “aim” for either deadline.

You will be given several submissions to allow you to “save” your work and to reduce

the temptation to wait until the last second to hit “Submit”.

FINAL IMPORTANT POINT: Once you have started the quiz you must complete it in

the allotted time, without exception. Do not close your browser: the clock will keep

running whether it is open or closed!

6 Introduction


L01-1


Name ___________________________ Date ____________ Partners_____________________________

Lab 1 – ELECTROSTATICS

OBJECTIVES

• To understand the difference between conducting and insulating

materials.

• To observe the effects of charge polarization in conductors and

insulators

• To understand and demonstrate two ways to charge an object:

conduction and induction

• To determine the polarity of charge on a charged object based on

macroscopic observations

OVERVIEW

Source: Paul M .Fishbane, Stephen Gasiorowicz, and Stephen T. Thornton. Physics for Scientists and

Engineers, 3rd

Edition. Prentice-Hall, Inc. Upper Saddle River, NJ. 2005. pp. 609-617

Electrostatics1 is the study of charges which are not in motion and the

interactions between them. Most of the phenomena we observe in the

study of electrostatics arise from ionization. Although atoms are

electrically neutral, their outermost electrons are sometimes easily

removed; when an atom gains or loses electrons, the resulting

imbalance of charge is referred to as ionization. Positive ions are

atoms which have lost electrons, while negative ions are atoms which

have gained electrons. A material with easily detached electrons

(which can then move through the material somewhat freely) is

referred to as a conductor. Conversely, a material with strongly

attached electrons is called an insulator. Objects can be charged by

making contact with another charged object, a phenomena known as

charging by conduction.

When objects are charged without coming into contact with a charge

source, the process is known as charging by induction; this process

primarily works with conductors. One method of charging involves

moving a charged object to the vicinity of two uncharged conductors

in contact with each other, as shown in Figure 1. An induced charge

flows to one conductor, leaving the other conductor oppositely

charged. When separated the conductors have equal, but opposite,

charges. As always, it is important to remember for these experiments

that charge is conserved and that like charges repel while opposite

charges attract.

1 Adapted from Dr. Richard Lindgren, Charlene Wyrick, Karyn Traphagen, and Lynn

Lucarto. The Shocking Truth: Lessons in Electrostatics. 2000.

L01-2 Lab 1 - Electrostatics


INTRODUCTION TO THE APPARATUS

The electroscope is an instrument that detects the presence of charge

on an object, either through actual contact (conduction) or through

induction. When the electroscope itself is charged, its two conductive

components (which vary from electroscope to electroscope) will

acquire like charge and deflect from the vertical position of

gravitational equilibrium. Thus a rod is proven to possess a charge

when contact between the rod and the electroscope transfers charge to

the previously neutral electroscope. A charged object brought in the

vicinity of the charged electroscope will change the angle of

deflection, indicating the presence of charge via induction (or induced

polarization). This process is explained in greater detail in the

experiments.

Figure 2: UVa Electroscope

Insulating

base Metal

Tube

Brass

Support

Steel

Figure 1: Charging by Induction

Step Step

Step Step

Lab 1 - Electrostatics L01-3


The UVa electroscopes have been designed with the following

considerations:

• The base of the electroscope is constructed out of acrylic, an insulator,

to minimize charge leakage to the table (or to the object on which the

base sits).

• The tube is made of copper, a conductor, and is suspended slightly

above its center of gravity so that it will quickly reach stable

equilibrium when no charge is present.

• The support structure is made of brass, another conductor. The

rounded edges of the support minimize charge leakage into the

atmosphere. [Sharp points on conductors tend to leak charge away

easily.] If a charged rod is brought into contact with the upper lip of

the support structure, charge will distribute across the brass and the

copper, causing the tube to deflect from the brass structure.

The following hints will optimize your use of the electroscope:

• Insulators (e.g. Teflon and acrylic) do not transfer charge easily to

other objects, so draw or scrape the rod across the brass lip in order to

transfer charge to the electroscope.

• Oils and salt transferred from your hand to the rod may adversely

affect experimental results. Only handle the unmarked end of the

Teflon rod and charge the marked end so as to minimize these effects.

[A useful mnemonic is to think of the marked end as “red hot”.]

• Occasionally you will have to touch the marked end to “bleed off” the

charge. To do this, grab the rod a couple of times along its length with

a clean hand (rubbing may actually charge, not discharge, the rod).

You may need to periodically clean the rod with alcohol.

• When using the electroscope to detect the presence of charge, it may

be necessary to bring a charged rod near the tube in a back-and-forth

rhythmic motion to cause visible movement of the tube.

• Humidity increases the charge leakage into the atmosphere. Ideally,

humidity should be below 50%. Winter is thus the best time in

Virginia for static electricity demonstrations, but we use air

conditioners and dehumidifiers to help improve the situation.

THE TRIBOELECTRIC SERIES

Materials possess various tendencies to acquire or to lose electrons; the

ordering of these tendencies is referred to as the triboelectric series.

When you use a silk cloth to charge a Teflon rod, you are engaging in

a process known as triboelectric charging. Teflon’s electrical nature

dictates that it will acquire a negative net charge, as it has a tendency

to take electrons from the silk. Glass, on the other hand, has a

tendency to acquire a positive net charge from silk. The list below



orders a number of common materials by their electrical nature2. Note

that human hands have a strong tendency to gain positive charge. We

will use Teflon and silk.

Human hands

Asbestos

Rabbit Fur

Glass, Mica

Human Hair

Nylon, Wool

Lead

Silk

Aluminum

Paper

Cotton

Steel Wood

Amber

Hard Rubber

Mylar

Nickel, Copper

Silver, Brass

Gold, Platinum

Polyester, Celluloid

Saran Wrap

Polyurethane

Polyethylene Polypropylene

Vinyl, Silicon

Teflon

Silicon Rubber

Figure 3

Question 1: If you rub a glass rod with silk, what is the polarity of

the charge on each object? What about Teflon and silk? Gold and

lead? Use the Triboelectric series above.

Glass and silk Glass _________ Silk __________

Teflon and silk: Teflon _________ Silk __________

Gold and lead: Gold _________ Lead __________

2 This Triboelectric series was adapted from Allen, Ryne C., Desco Industries Inc. (DII),

December 2000. Downloaded from

http://www.esdjournal.com/techpapr/ryne/ryntribo.doc, July 31, 2002.

Tendency to gain

POSITIVE charge

Tendency to gain

NEGATIVE charge



INVESTIGATION 1: CONDUCTORS AND INSULATORS

The purpose of the experiment is to observe the electrical

properties associated with an insulator and a conductor. Recall

that, as stated in the Overview, electrons can move freely through

conductors, while they are not free to move within insulators.

The following equipment is needed for this investigation:

• Spinner • Electroscope

• Silk cloth • Teflon rods (2)

• Brass rod • Wooden rod

• Acrylic rod

Activity 1-1: Teflon

In the following activity, you will observe the interaction between

two negatively charged Teflon rods, one held in your hand and one

placed on a spinner. You will charge only one end of each Teflon

rod.

The “spinner” consists of a metal pin attached to a plastic acrylic

base and a second piece of plastic which rotates on the metal pin. It

thus rotates very freely. The rod is positioned on this second piece

of plastic.

Prediction 1-1: Consider Figure 4. Predict what will happen if

you position the charged end of the Teflon rod [in your hand]

alongside and parallel to the charged end of the Teflon rod on the

spinner without letting them touch? What about along the

uncharged end of the Teflon rod on the spinner? Do not yet try it.

Figure 4



Question 1-1: How is it that we can charge one end of the Teflon

rod and leave the other end uncharged? Is this true for all

materials?

1. Charge one end of the first Teflon rod by striking or rubbing it

on the silk and place this Teflon rod on the spinner. Now

charge one end of the second Teflon rod.

2. Use the spinner and the Teflon rods to test your predictions:

After charging both rods, hold one Teflon rod in your hand

parallel to the Teflon rod on the spinner as shown in Figure 4 in

order to ensure the greatest possible interaction between the

two. If nothing happens, you may need to recharge the Teflon

rods.

Question 1-2: What were your results? In particular, was there

any interaction with the uncharged end of the rod on the spinner

(you need to hold the rods very close together to see)? Discuss.

The Charging Process: When you strike the silk with the Teflon rod, a charge transfer

occurs between the two materials. As Teflon attracts negative

charge, the Teflon rod attracts the loose electrons from the silk’s

surface and becomes negatively charged. Because charge is

conserved, the silk is left positively charged. Transfer of electrons

is responsible for charging; atoms do not transfer protons.



Activity 1-2: Movement of Charge in Metals

Figure 5

1. Use the charged Teflon rod to charge the electroscope by

rubbing the top of the brass support with the end of the Teflon

rod that is charged. You may need to charge the rod and rub

the support a few times until the tube deflects a little.

2. Recharge the Teflon rod and add more charge to the

electroscope.

Question 1-3: What did you observe? Discuss your observations

in terms of metal’s conducting properties and the charge

distribution.

How the Electroscope Works: Metals contain some electrons

which are not tightly bound to the atoms and are free to move about

through the conductor. When you charge the electroscope, excess

charge distributes itself across the conducting parts of the

electroscope (the brass support and the metal tube). This

distribution occurs as a result of the force of repulsion between like

charges, which attempt to move as far away from each other as

possible. After the charges reach static equilibrium, the tube should

be deflected at a constant angle.

Insulating Metal

Tube

Brass

Support



Activity 1-3: Using Conductors to Discharge the Electroscope

When you provide the electrons in a charged object a route of

escape through contact with an uncharged conducting object,

electrons will flow from the charged object to the uncharged

conducting object until electrical balance is achieved.

Prediction 1-2: Predict what would happen if you touched the top

of the charged electroscope with your finger? What does this

indicate about the conducting properties of the human body?

1. Observe as you touch the top of the brass support with your

finger and record the results:

Question 1-4: Were your observations consistent with your

prediction? Discuss.

Prediction 1-3: What would happen if you touched the top of the

charged electroscope with a brass rod held in your hand? An

acrylic rod? Justify your prediction on the basis of the properties

of conductors and insulators.

When you bring the metal parts of the charged electroscope into

contact with a conducting element, you provide a pathway for the

charges to flow in to (or out of) the electroscope so as to balance

the forces between the charges, leaving it electrically neutral. This

is called “grounding” or “discharging”.

Touching a charged object with an insulator will not completely

discharge it. Some charges may transfer near the region of contact,

but, as charges are not free to move about in an insulator, most of

the charge imbalance will remain.



2. Charge the electroscope. Touch the charged electroscope with

the (uncharged) brass rod. Re-charge the electroscope (if

necessary) and touch it with the (uncharged) acrylic rod.

Observe and record the results.

Question 1-5: Explain your observations in terms of each rod’s

conducting properties.

Prediction 1-4: Is wood a conductor or an insulator? What could

you do to see?

3. Re-charge the electroscope (if necessary). Touch the charged

electroscope with the wooden rod and observe and record the

results.

Question 1-6: Is wood a conductor or an insulator? Explain in

light of your observations.



INVESTIGATION 2: CHARGE POLARIZATION

An object is said to be neutral if it contains the same number of

positive and negative charges. A neutral object can, however,

produce some of the same phenomena as a charged object as a

result of a process known as polarization. We already know that

opposite charges attract. If we recall that charges are somewhat

free to move within an object, we should not be surprised that a

positively charged object will induce a charge alignment in a

neutral object so that the object’s electrons are as near to the

positively charged object as possible. As a result, the neutral

object will appear to react to an electric force as though it were

charged. When you charge by induction, you are exploiting

polarization.

Recall that in a conductor, electrons are free to move. Polarization

in a conductor, then, is a result of a movement of electrons to one

side of an object. Electrons are not free to move (at least not very

far) in an insulator. This does not mean, however, that an insulator

does not experience polarization. Polarization in an insulator is a

result of an alignment of the charge within each individual atom or

molecule.

Figure 6

A negative charge in the vicinity of an atom, for example, will

repel the atom’s electron cloud and attract the positive nucleus.

This results in a rearrangement of the charges of the atom

(Figure 6). The closer the atom is to the charged object, the more

it will be “stretched”. Polarization does not create a permanent

charge; it is instead a temporary effect caused by the proximity of a

charged object. The polarized object acquires no net charge.

Polarized atom Charged object



Figure 7

Prediction 2-1: What charges will the left and right sides of the

insulating rod tend to have in Figure 7? Using Figure 6 as a guide,

indicate on Figure 7 how the charges are distributed. Discuss.

Activity 2-1: Charge Polarization in an Insulator

1. Place the acrylic rod on the spinner.

Prediction 2-2: Predict what will happen if you bring a charged

Teflon rod near the uncharged acrylic rod? Will it matter which

end of the acrylic rod you place the charged rod near? Will it

matter which side of the acrylic rod you choose?

Insulating rod

Charged rod



2. Test your predictions and record your observations:

Question 2-1: Discuss the agreement between your prediction and

your observations.

Activity 2-2: Charge Polarization in a Conductor

1. Use the charged Teflon rod to charge the electroscope.

2. Recharge the Teflon rod and bring the charged side (not the

end, as the simplistic 2-D rendering shown in Figure 8 might

imply) of the rod slowly near the lower end of the tube. At

first you should notice repulsion between the tube and the

charged Teflon rod; as the rod slowly approaches the tube, it

should then begin to attract.

Question 2-2: In terms of polarization, what causes the attraction

of the tube to the Teflon rod? Why does the tube initially repel and

then attract? Discuss.

1. Charge the electroscope using the Teflon rod and silk.

2. Bring the charged Teflon rod near the bottom of the brass

support (on the side opposite from the tube, see Figure 8

(below), but not touching. You should observe an increased

deflection angle between the tube and the brass support.



Question 2-3: In terms of polarization, what causes this increased

deflection?

Figure 8

Question 2-4: What charge sign do the elements of the

electroscope have? Indicate in the diagram above the sign of any

charged element of the electroscope. Discuss.

INVESTIGATION 3: CHARGING BY INDUCTION

When you charge the electroscope by touching the negatively

charged rod to the top of the brass support, you are charging via

conduction, which requires contact. The electroscope acquires the

same charge as the charged rod; the negatively charged rod (which

has taken electrons from the silk) distributes electrons on the

electroscope. Charging via conduction necessitates that the

charged object physically touch the object to be charged, and both

objects will have the same sign of charge.

Charging via induction occurs without contact between the

charged object and the object to be charged. The charged object



(which we will refer to as an inducer) is brought near the neutral

object, inducing polarization. Suppose the inducer has a negative

charge. If you place your finger on the polarized object, your

finger will drain electrons from the object, thus creating a net

positive charge (Figure 9). When your finger and then the inducer

are both removed, this net positive charge remains. The newly

charged object is thus of the opposite polarity as the initially

charged inducer.

Figure 9

You do not need any new equipment for this investigation.

Activity 3-1: Charging by Induction

1. Ground the electroscope to neutralize it. You can do this by

touching the top of the brass support with your finger, thereby

providing a pathway for the charge to the ground.

2. Have one partner hold his/her finger to the top of the brass

support.

3. While the finger is on the brass support, have another partner

bring the side (not the tip, as the figure seems to indicate) of the

charged Teflon rod to the base (on the side away from the tube,

as shown) of the brass support (without touching it) until you

see a deflection of the tube.

4. Remove the finger from the support, and then remove the

Teflon rod.



5. Record your observations:

Question 3-1: What is the sign of the charge on the electroscope?

Explain what has happened.

Question 3-2: How can you verify the sign of charge on the

electroscope? Describe. Now try it, and discuss your results.



L02-1


Name ____________________________ Date __________ Partners______________________________

Lab 2 – GAUSS’ LAW

On all questions, work together as a group.

1. The statement of Gauss’ Law:

(a) in words:

The electric flux through a closed surface is equal to the total charge

enclosed by the surface divided by 1.

(b) in symbols:

2. The next few questions involve point charges.

(a) Draw the electric field lines in the vicinity of a positive charge +Q.

(b) Do the same for a negative charge -Q.

3. Consider a “Gaussian sphere”, outside of which a charge +Q lies. Remember, a

Gaussian surface is just a mathematical construct to help us calculate electric fields.

Nothing is actually there to interfere with any electric charges or electric fields.

1 ε0 = 8.85 × 10-12 C2/N·m2

• + Q

• – Q

• + Q

L02-2 Lab 2 – Gauss’ Law


(a) Draw several electric field lines from +Q, but only ones that intersect the sphere.

(For this question, omit field lines that don't intersect the sphere. This is to keep

the drawing looking neat.)

(b) How much charge is enclosed by the sphere? Applying Gauss’ Law, what is the

total electric field flux through the sphere? Justify your answer.

(c) Looking back at your drawing, field lines impinge on the spherical surface from

the outside heading inward (this is defined as negative flux) and eventually

impinge on another part of the surface from the inside heading outward (positive

flux). Does it seem reasonable that the total flux through the sphere is exactly

zero? Explain why the total flux through the sphere is exactly zero when the field

lines exit through a larger area!

(d) Suppose we replaced the sphere with a cube. Would the total flux still be zero?

Explain.

Note that it would be difficult to actually calculate the electric field flux through these

surfaces (although you could certainly do it) because the electric field strength and

angle of intersection vary over the surface. Applying Gauss’ Law, however, made it

easy.

Lab 2 – Gauss’ Law L02-3


4. Now consider a Gaussian sphere centered on +Q.

(a) Draw some electric field lines. Make them long enough to intersect the sphere.

(b) Is the total electric field flux through the sphere positive or negative? Does this

make sense, considering the charge enclosed? Discuss.

(c) In symbols, what is the total flux through the sphere? [Use equation in step 1(b).]

Notice that in this case the electric field lines intersect the sphere perpendicular to its

surface and that the electric field strength is uniform over the surface.

(d) How do we know that the electric field strength does not vary over the surface?

• +Q



(e) Because of the simplifying conditions discussed in part (d), we can apply Gauss’

Law to find the electric field due to +Q. Find the electric field for a point charge

+Q. (Recall that the surface area of a sphere is 24 rπ .)

(f) Graph E (the magnitude of E) versus r, using the axes given. [Both axes have

linear scales.]

(g) If we had picked a cube as our Gaussian surface instead of a sphere, would it still

have been easy to determine the total flux through the surface? What about

calculating the electric field strength? Explain.

5. The previous problem was mathematically fairly simple. Here’s another problem

requiring Gauss’ Law, but this time you will have to do a bit of integration.

NOTE: Keep your results in symbolic form and only substitute in

numbers when asked for a numerical result. Also, pay careful attention to

the distinction between the radius of the sphere, R, and the distance, r,

from the center of the sphere at which you are evaluating E.

Consider a small sphere (an actual sphere, not a Gaussian surface) of radius R = 0.1 m

that is charged throughout its interior, but not uniformly so. The charge density

is Brρ = , where r is the distance from the center, and B = 10-4

C/m4 is a constant. Of

course, for r greater than R, the charge density is zero.

R = 0.1 m

E

r



(a) What does the charge density converge to as you approach the center of the

sphere? Does it increase or decrease as we move toward the surface? Explain.

(b) What does E converge to as you approach the center of the sphere? How do you

know? How does this compare to the E of a point charge? [Hint: Consider the

symmetry of the problem.]

(c) Apply Gauss’ Law to find an expression for E when r is less than R. [Hint: the

volume of a thin spherical shell of radius r and thickness dr is 24dV r drπ= .]

(d) What is E at r = 0.05 m? [Your answer should be in N/C.]



(e) Apply Gauss’ Law to find an expression for E when r is greater than R.

(f) What is E at r = 5.0 m?

(g) Make an approximate graph of E versus r on the axes shown below:

(h) Determine the total charge inside the sphere. Outside the sphere, how does E

compare to E of a point charge of that magnitude? Verify for r = 5.0 m.

(i) On your graph in part (g) above, use a dotted line to represent the electric field if

the sphere shrank to a point charge but still contained the same total charge.

Emax

r 0.1 m 0

0



6. Consider two point charges of opposite sign as shown:

(a) In what direction is the electric field at the point indicated? Draw a vector on the

sketch above at the point X to represent E. [Hint: Use vector addition.]

(b) On the sketch below, draw several (at least ten) electric field lines for the

configuration above. This should be enough to indicate E in much of the vicinity

of the two charges.

(c) On your drawing above, add a spherical Gaussian surface that encloses both

charges, centered on the point midway between the charges. Make it fairly large,

but be sure several of the electric field lines penetrate the Gaussian sphere. Add

more field lines as needed.

+ Q • • – Q

+ Q • • – Q

X



(d) Look carefully at your drawing above. In what direction does E point (inward?

outward? tangent?) at various locations on the Gaussian surface? Redraw your

Gaussian surface below and draw short arrows on the surface indicating the

direction of E on the Gaussian surface.

(e) Consider the following argument from a student who is trying to determine E

somewhere on the previous Gaussian surface:

“The total charge enclosed by the surface is zero.

According to Gauss’ law this means the total electric field

flux through the surface is zero. Therefore, the electric field

is zero everywhere on the surface.”

Which, if any, of the three sentences are correct?

Explain how the student came to an incorrect conclusion.

+ Q • • – Q



(f) Refer back to 6(a). If you were asked to calculate the electric field at that point,

would you attempt to apply Gauss’ Law or would you use another method?

Discuss.

Gauss’ law is always true, no matter how complicated the distribution of electric charges.

In fact it’s even true when the charges are in motion. However, it’s rare to find a

situation with enough symmetry that applying Gauss’ law becomes a convenient method

to calculate the electric field.

Electric fields in the vicinity of a conductor

A point charge +Q lies at the center of an uncharged, hollow, conducting spherical shell

of inner radius Rin and outer radius Rout as shown. Your ultimate goal is to find the electric

field at all locations for this arrangement.

7. Why is it important to know whether the shell is a conductor or insulator? Is there

any real difference?

Rin

Rout

+Q •



8. What does the charge distribution within the shell itself (between Rin and Rout ) look

like

(a) before the point charge +Q is inserted? (draw a picture)?

(b) with the point charge +Q in place? (draw a picture)?

(c) Are the charge distributions the same? Why or why not?

9. What is the electric field within the shell itself (between Rin and Rout , NOT including

the surfaces)

(a) before the point charge +Q is inserted?



(b) after the point charge +Q is inserted?

(c) How did the fact that the shell is a conductor help you answer the two previous

questions?

10. We are still considering the conducting shell. Draw electric field lines in the region

0 < r < Rin (r is the distance from the point charge) and for the region r > Rout. What

are the field lines between Rin and Rout? Also show the relative amount of charge that

has moved to the inner and outer surfaces of the conductor.

11. Use Gauss’ law to find the electric field as a function of r for all three regions, and

then graph it below.

E

r

+Q •



12. Compare the electric field for r just inside Rin and just outside Rout. Are the

magnitudes equal? Explain.

13. In specifying the electric field, do you need to give any other information besides the

distance from the point charge (for instance, whether you're on the right side or the

left side of the point charge, or something like that)? Discuss.

L03-1

University of Virginia Physics Department Modified from P. Laws, D. Sokoloff, R. Thornton PHYS 2419, Fall 2011

Name __________________________ Date ____________ Partners______________________________

Lab 3 - SIMPLE DC CIRCUITS

OBJECTIVES

• To understand how a potential difference (voltage) can

cause an electric current through a conductor.

• To learn to design and construct simple circuits using

batteries, bulbs, wires, and switches.

• To learn to draw circuit diagrams using symbols.

• To understand currents at all points in simple circuits.

• To understand the meaning of series and parallel

connections in an electric circuit and how current flows

through them.

OVERVIEW

In this lab* you are going to consider theories about electric

charge and potential difference (voltage) and apply them to

electric circuits.

A battery is a device that generates an electric potential

difference (voltage) from other forms of energy. An ideal

battery will maintain a constant voltage no matter what is

connected to it. The batteries you will use in these labs are

known as chemical batteries because they convert internal

chemical energy into electrical energy.

As a result of a potential difference, electric charge is repelled

from one terminal of the battery and attracted to the other.

However, no charge can flow out of a battery unless there is a

conducting material connected between its terminals. If this

conductor happens to be the filament in a small light bulb, the

flow of charge will cause the light bulb to glow.

* Some of the activities in this lab have been adapted from those designed by the Physics Education Group

at the University of Washington

L03-2 Lab 3 - Simple DC Circuits


You are going to see how charge flows in wires and bulbs

when energy has been transferred to it by a battery. You will

be asked to develop and explain some models that predict how

the charge flows. You will also be asked to devise ways to test

your models using current and voltage probes, which can

measure the rate of flow of electric charge (current) through a

circuit element and the potential difference (voltage) across a

circuit element, respectively, and display these quantities on a

computer screen.

Then you will examine more complicated circuits than a single

bulb connected to a single battery. You will compare the

currents through different parts of these circuits by comparing

the brightness of the bulbs, and also by measuring the currents

using current probes.

The following figure shows the parts of the bulb, some of

which may be hidden from view.

Non-conducting

ceramic material

Conducting

metallic

material

Filament

Figure 1-1: Diagram of wiring inside a light bulb.

NOTE: These bulbs do NOT obey “Ohm’s Law” in that the

voltage across the bulb is not simply proportional to the

current through it. However, both the voltage across the bulb

and the bulb’s brightness are monotonically increasing

functions of the current through the bulb. In other words,

“more current means more voltage” and “more current means

brighter”.

Prediction 1-1: In Figure 1-2 (below) are shown several

models that people often propose. Which model do you think

best describes the current through the bulb? Explain your

reasoning.

Lab 3 - Simple DC Circuits L03-3


Model A: There is an electric current from

the top terminal of the battery to the bulb

through wire 1, but no current back to the

base of the battery through wire 2, since the

current is used up lighting the bulb.

Model B: There is an electric current in

both wires 1 and 2 in a direction from the

battery to the bulb.

1

2

Model C: The electric current is in the

direction shown, but there is less current in

the return wire (wire 2), since some of the

current is used up lighting the bulb.

2

1

1

2

Model D: The electric current is in the

direction shown, and the magnitude of the

current is the same in both wires 1 and 2.

1

2

Figure 1-2: Four alternative models for current

For the Investigations in this lab, you will need the following:

• three current probes • two voltage probes

• three bulbs (#14) and holders • D cell battery

• momentary contact switch • knife switch

• nine wires with alligator clips • battery holder

The current probe is a device that measures current and

displays it as a function of time on the computer screen. It will

allow you to explore the current at different locations and

under different conditions in your electric circuits.

To measure the current through an element of the circuit, you

must break open the circuit at the point where you want to

measure the current, and insert the current probe. That is,

disconnect the circuit, put in the current probe, and reconnect

with the probe in place.

NOTE: The current probe measures both the magnitude and

the direction of the current. A current in through the “+”

terminal and out through the “–” terminal (in the direction of

the arrow) will be displayed as a positive current. Thus, if the

current measured by the probe is positive, you know that the

current must be counterclockwise in Figure 1-3 from the “+”

terminal of the battery, through the bulb, through the switch,

and toward the “–” terminal of the battery. On the other hand,

if the probe measures a negative current, then the current must

be clockwise in Figure 1-3 (into the “–” terminal and out of the

“+” terminal of the probe).



+ –

Current Probe

–

+

Interface

Figure 1-3

Figure 1-3 shows a circuit with a battery, bulb, switch, and

current probe connected to the computer interface.

Figure 1-4(a) below, shows a simplified diagram.

CPB

CPA CPA

(a) (b)

+ +

- -

– + – +

Figure 1-4

Look at Figure 1-4(b) and convince yourself that if the currents

measured by current probes CPA and CPB are both positive,

this shows that the current is in a counterclockwise direction

around all parts of the circuit.

INVESTIGATION 1: MODELS DESCRIBING CURRENT

Activity 1-1: Developing a Model for Current in a Circuit

1. Be sure that current probes CPA and CPB are plugged into

the interface.

2. In DataStudio, open the experiment file called

L03A1-1 Current Model. Current for two probes versus

time should appear on the screen. The top axes display the

current through CPA and the bottom the current through

CPB. The amount of current through each probe is also

displayed digitally on the screen.

3. To begin, set up the circuit in Figure 1-4(b). Use the

“momentary contact” switch, not the “knife” switch. Begin



graphing, and try closing the switch for a couple of

seconds and then opening it for a couple of seconds.

Repeat this a few times during the time when you are

graphing.

4. Print one set of graphs for your group.

NOTE: You should observe carefully whether the current

through both probes is essentially the same or if there is a

significant difference (more than a few percent). Write your

observation:

Question 1-1: You will notice after closing the switch that the

current through the circuit is not constant in time. This is

because the electrical resistance of a light bulb changes as it

heats up, quickly reaching a steady-state condition. When is

the current through the bulb the largest – just after the switch

has been closed, or when the bulb reaches equilibrium? About

how long does it take for the bulb to reach equilibrium?

Question 1-2: Based on your observations, which of the four

models in Figure 1-2 seems to correctly describe the behavior

of the current in your circuit? Explain based on your

observations. Is the current “used up” by the bulb?



INVESTIGATION 2: CURRENT AND POTENTIAL DIFFERENCE

Switch

Bulb Wire

Battery

+

-

+

-

Figure 2-1: Some common circuit symbols

Using these symbols, the circuit with a switch, bulb, wires, and

battery can be sketched as on the right in Figure 2-2.

Figure 2-2: A circuit sketch and corresponding circuit diagram

There are two important quantities to consider in describing the

operation of electric circuits. One is current, which is the flow

of charges (usually electrons) through circuit elements. The

other is potential difference, often referred to as voltage. Let's

actually measure both current and voltage in a familiar circuit.

NOTE: The voltage probe measures both the magnitude and

the polarity of the voltage. A very common practice is to is to

label wires with color (a “color code”). For our voltage probes,

when the red wire is more positive than the black wire, the

measured voltage difference will be positive. Conversely,

when the black wire is more positive than the red wire, the

measured voltage difference will be negative.

Figure 2-3(a) shows our simple circuit with voltage probes

connected to measure the voltage across the battery and the

voltage across the bulb. The circuit is drawn again

symbolically in Figure 2-3(b). Note that the word across is

very descriptive of how the voltage probes are connected.

Activity 2-1: Measuring Potential Difference (Voltage)

1. To set up the voltage probes, first unplug the current probes

from the interface and plug in the voltage probes.



2. Open the experiment file called L03A2-1 Two Voltages to

display graphs for two voltage probes as a function of time.

3. Connect the circuit shown in Figure 2-3.

+

-

VPA +

-

VPB

+

-

VPA

+ -

VPB (a)

(b)

Figure 2-3: Two voltage probes connected to measure the voltages across the battery and the bulb.

Prediction 2-1: In the circuit in Figure 2-3, how would you

expect the voltage across the battery to compare to the voltage

across the bulb with the switch open and with the switch

closed? Explain.

4. Now test your prediction. Connect the voltage probes to

measure the voltage across the battery and the voltage

across the bulb simultaneously.

5. Click on Start, and close and open the switch a few times.


Question 2-1: Did your observations agree with your

Prediction 2-1? Discuss.



Question 2-2: Does the voltage across the battery change as

the switch is opened and closed? What is the “open circuit”

battery voltage, and what is the battery voltage with a “load”

on it (i.e. when it’s powering the light bulb)?

Activity 2-2: Measuring Potential Difference (Voltage) and

Current

1. Connect a voltage and a current probe so that you are

measuring the voltage across the battery and the current

through the battery at the same time. (See Figure 2-5.)

2. Open the experiment file called L03A2-2 Current and

Voltage to display the current CPB and voltage VPA as a

function of time.

CPB

+ - +

-

VPA

+

-

VPA

+ -

CPB

Figure 2-5: Probes connected to measure the voltage across

the battery and the current through it.

3. Click on Start, and close and open the switch a few times,

as before.

Question 2-3: Explain the appearance of your current and

voltage graphs. What happens to the current through the

battery as the switch is closed and opened? What happens to

the voltage across the battery?



4. Find the voltage across and the current through the battery

when the switch is closed, the bulb is lit, and the values are

constant. Use the Smart Tool and/or the Statistics feature.

Average voltage: ____________

Average current: ____________

Prediction 2-2: Now suppose you connect a second bulb in

the circuit, as shown in Figure 2-6. How do you think the

voltage across the battery will compare to that with only one

bulb? Will it change significantly? What about the current in

the circuit and the brightness of the bulbs? Explain.

Comment: These activities assume identical bulbs.

Differences in brightness may arise if the bulbs are not exactly

identical. To determine whether a difference in brightness is

caused by a difference in the currents through the bulbs or by a

difference in the bulbs, you should exchange the bulbs.

Sometimes a bulb will not light noticeably, even if there is a

small but significant current through it. If a bulb is really off,

that is, if there is no current through it, then unscrewing the

bulb will not affect the rest of the circuit. To verify whether a

non-glowing bulb actually has a current through it, unscrew the

bulb and see if anything else in the circuit changes.

5. Connect the circuit with two bulbs, and test your

prediction. Take data. Again measure the voltage across

and the current through the battery with the switch closed.

Average voltage:__________

Average current:__________




VPA

VPA

CPB

CPB

Figure 2-6: Two bulbs connected with voltage and current probes.

Question 2-4: Did the current through the battery change

significantly when you added the second bulb to the circuit (by

more than, say, 20%)?

Question 2-5: Did the voltage across the battery change

significantly when you added the second bulb to the circuit (by

more than 20% or so)?

Question 2-6: Does the battery appear to be a source of

constant current, constant voltage, or neither when different

elements are added to a circuit?

Comment: A chemical battery is a fair approximation to an

ideal voltage source when it is fresh and when current demands

are small. Usage and age causes the battery’s internal

resistance to increase and when this resistance becomes

comparable to that of other elements in the circuit, the battery’s

voltage will sag noticeably.



INVESTIGATION 3: CURRENT IN SERIES CIRCUITS

In the next series of activities you will be asked to make a

number of predictions about the current in various circuits and

then to compare your predictions with actual observations.

Whenever your experimental observations disagree with your

predictions you should try to develop new concepts about how

circuits with batteries and bulbs actually work.

Helpful symbols: > “is greater than”, < “is less than”, = “is

equal to”. For example, B>C>A

Prediction 3-1: What would you predict about the relative

amount of current going through each bulb in Figures 3-1 (a)

and (b)? Write down your predicted order of the amount of

current passing through bulbs A, B and C.

Activity 3-1: Current in a Simple Circuit with Bulbs

We continue to see which model in Figure 1-2 accurately

represents what is happening. You can test your Prediction 3-1

by using current probes.

CPA - +

CPB - +

CPA - +

CPB - + A

B

C

(a) (b)

Figure 3-1

Figure 3-1 shows current probes connected to measure the

current through bulbs. In circuit (a), CPA measures the current

into bulb A, and CPB measures the current out of bulb A. In

circuit (b), CPA measures the current into bulb B while CPB

measures the current out of bulb B and the current into bulb C.

Spend some time and convince yourself that the current probes

do indeed measure these currents.

1. Open the experiment file L03A3-1 Two Currents to

display the two sets of current axes versus time.

2. Connect circuit (a) in Figure 3-1.



3. Begin graphing. Close the switch for a second or so.

Open it for a second or so, and then close it again.

4. Use the Smart Tool to measure the currents into and out of

bulb A when the switch is closed:

Current into bulb A:______

Current out of bulb A:______

Question 3-1: Are the currents into and out of bulb A equal,

or is one significantly larger (do they differ by more than a few

percent)? What can you say about the directions of the

currents? Is this what you expected?

5. Connect circuit (b) in Figure 3-1. Begin graphing current

as above, and record the measured values of the currents.

Current through bulb B:_____

Current through bulb C:_____


Question 3-2: Consider your observation of the circuit in

Figure 3-1b with bulbs B and C in it. Is current “used up” in the

first bulb, or is it the same in both bulbs?

Question 3-3: Is the ranking of the currents in bulbs A, B and C

what you predicted? Discuss.



Question 3-4: Based on your observations, how is the

brightness of a bulb related to the current through it?

Question 3-5: Formulate a qualitative rule (in words, not an

equation) for predicting whether current increases or decreases as

the total resistance of the circuit is increased.

Comment: The rule you have formulated based on your

observations with bulbs may be qualitatively correct – correctly

predicting an increase or decrease in current – but it won't be

quantitatively correct. That is, it won’t allow you to predict the

exact sizes of the currents correctly. This is because the

electrical resistance of a bulb changes as the current through

the bulb changes.

INVESTIGATION 4: CURRENT IN PARALLEL CIRCUITS

There are two basic ways to connect resistors, bulbs or other

elements in a circuit – series and parallel. So far you have

been connecting bulbs and resistors in series. To make

predictions involving more complicated circuits we need to

have a more precise definition of series and parallel. These are

summarized in the box below.



Series connection: Two resistors or bulbs are in series if

they are connected so that the same

current that passes through one resistor

or bulb passes through the other. That

is, there is only one path available for

the current.

Series

Parallel connection: Two resistors or bulbs are in parallel if

their terminals are connected together such

that at each junction one end of a resistor

or bulb is directly connected to one end of

the other resistor or bulb, e.g., junction 1 in the diagram.

Similarly, the other ends are connected

together (junction 2). Parallel

junction 2

junction 1

i2 i1

i

i

i

It is important to keep in mind that in more complex circuits,

say with three or more elements, not every element is

necessarily connected in series or parallel with other elements.

Let’s compare the behavior of a circuit with two bulbs wired in

parallel to the circuit with a single bulb.

(a) (b)

A D E

Figure 4-1

Figure 4-1 shows two different circuits: (a) a single bulb circuit

and (b) a circuit with two bulbs identical to the one in (a)

connected in parallel to each other and in parallel to the

battery.



Prediction 4-1: What do you predict about the relative amount

of current through each bulb in a parallel connection, i.e.,

compare the current through bulbs D and E in Figure 4-1 (b)?

Note that if bulbs A, D and E are identical, then the circuit in

Figure 4-2 is equivalent to circuit 4-1(a) when the switch S is

open (as shown) and equivalent to circuit 4-1(b) when the

switch S is closed.

D E

S

Figure 4-2

When the switch is open, only bulb D is connected to the

battery. When the switch is closed, bulbs D and E are

connected in parallel to each other and in parallel to the battery.

Prediction 4-2: How do you think that closing the switch in

Figure 4-2 affects the current through bulb D?

Activity 4-1: Current in Parallel Branches

You can test Predictions 4-1 and 4-2 by connecting current

probes to measure the currents through bulbs D and E.

1. Continue to use the experiment file called L03A3-1 0Two

Currents. Clear any old data.

2. Connect the circuit shown below in Figure 4-3. Use the

momentary contact switch for S1.



NOTE: The purpose of switch S1 is to “save the battery”.

It is to be closed when taking data but open at all other times.

We use the momentary contact switch as it will “pop open”

when you let go.

S1 S2

D

CPA + –

E

CPB + –

Figure 4-3

3. Close switch S1 and begin graphing the currents through

both probes. Then close the switch S2 for a second or so,

open it for a second or so, and then close it again.

4. Open switch S1 to save the battery.


6. Use the Smart Tool to measure both currents.

Switch S2 open:

Current through bulb D: _____

Current through bulb E: _____

Switch S2 closed:

Current through bulb D: _____

Current through bulb E: _____

Question 4-1: Did closing the switch S2 and connecting bulb

E in parallel with bulb D significantly affect the current

through bulb D? How do you know? [Note: you are making a

very significant change in the circuit. Think about whether the

new current through D when the switch is closed reflects this.]

The voltage maintained by a battery doesn’t change

appreciably no matter what is connected to it (i.e. an ideal

battery is a constant voltage source). But what about the

current through the battery? Is it always the same no matter

what is connected to it, or does it change depending on the

circuit? This is what you will investigate next.



Prediction 4-3: What do you predict about the amount of

current through the battery in the parallel bulb circuit –

Figure 4-1 (b) – compared to that through the single bulb

circuit – Figure 4-1 (a)? Explain.

Activity 4-2: Current Through the Battery

1. Test your prediction with the circuit shown in Figure 4-4.

Open experiment file, L03A4-2 Three Currents.

S1 S2

D

CPA + –

E

CPB + –

CPC – +

Figure 4-4

Figure 4-4 shows current probes connected to measure the

current through the battery and the current through bulbs D and

E.

2. Insert a third current probe (CPC) as shown in Figure 4-4.

3. Close switch S1 and begin graphing while closing and

opening the switch S2 as before.

4. Open switch S1 to save the battery.

5. Print one set of graphs for your group

6. Label on your graphs when the switch S2 is open and when

it is closed. Remember that switch S1 is always closed

when taking data, but open when not in order to save the

battery.

7. Measure the currents through the battery and through the

bulbs:

Switch S2 open:

Current through battery:_____

Current through bulb D:____

Current through bulb E:____



Switch S2 closed:

Current through battery:_____

Current through bulb D:____

Current through bulb E:____

Question 4-2: Does the current through the battery change as

you predicted? If not, why not?

Question 4-3: Does the addition of more bulbs in parallel

increase, decrease or not change the total resistance of the

circuit?

INVESTIGATION 5: MORE COMPLEX SERIES AND PARALLEL CIRCUITS

Now you can apply your knowledge to some more complex

circuits. Consider the circuit consisting of a battery and two

identical bulbs, A and B, in series shown in Figure 5-1 (a).

B

A

B

A

C

(a) (b) Figure 5-1

What will happen if you add a third identical bulb, C, in

parallel with bulb B as shown in Figure 5-1 (b)? You should

be able to predict the relative brightness of A, B, and C based



on previous observations. An important tough question is: how

does the brightness of A change when C is connected in

parallel to B?

Question 5-1: In Figure 5-1 (b) is bulb A in series with bulb

B, with bulb C, or with a combination of bulbs B and C? (You

may want to go back to the definitions of series and parallel

connections.)

Question 5-2: In Figure 5-1 (b) are bulbs B and C connected

in series or in parallel with each other, or neither?

Question 5-3: Is the resistance of the combination A, B and C

in Figure 5-1 (b) larger than, smaller than or the same as the

combination of A and B in Figure 5-1 (a)?

Prediction 5-1: Predict how the current through bulb A will

change, if at all, when circuit 5-1 (a) is changed to 5-1 (b)

(when bulb C is added in parallel to bulb B). What will happen

to the brightness of bulb A? Explain the reasons for your

predictions.

Prediction 5-2: Predict how the current through bulb B will

change, if at all, when circuit 5-1 (a) is changed to 5-1 (b)

(when bulb C is added in parallel to bulb B). What will happen

to the brightness of bulb B? Explain the reasons for your

predictions. [This is difficult to do without a calculation, but at

least explain your considerations.]



Activity 5-1: A More Complex Circuit

1. Set up the circuit shown in Figure 5-2. Again, use the

momentary contact switch for S1 to save the battery.

2. Convince yourself that this circuit is identical to

Figure 5-1 (a) when the switch, S, is open, and to

Figure 5-1 (b) when the switch is closed.

3. Continue to use the experiment file L03A4-2 Three

Currents. Clear any old data.

S1

S2

B

CPB + –

C

CPC + –

CPA + –

A

Figure 5-2

4. Close the battery switch S1 and begin graphing. Observe

what happens to the current through bulb A (i.e. through

the battery) and the current through bulbs B and C when

the switch S2 to bulb C is opened and closed.

5. Open the battery switch S1.


7. Use the Smart Tool to find the following information:

Without bulb C in the circuit (S2 open):

current through A:___________

current through B:___________

current through C:___________

With bulb C in the circuit (S2 closed):

current through A:___________

current through B:___________

current through C:___________



Question 5-4: What happened to the current through bulbs A

and B as the switch to bulb C was opened and closed?

Compare to your predictions.

Question 5-5: What happens to the current through the battery

when bulb C is added into the circuit? What do you conclude

happens to the total resistance in the circuit?

WRAP-UP

Question 1: Consider your observations and discuss the

following statement: “In a series circuit, the current is the

same through all elements.”

Question 2: Consider your observations and discuss the

following statement: “The sum of the currents entering a

junction equals the sum of the currents leaving the junction.”



L04-1


Name ________________________ Date ____________ Partners_______________________________

Lab 4 – OHM’S LAW AND

KIRCHHOFF’S CIRCUIT RULES

AMPS

VOLTS

+

-

OBJECTIVES

• To learn to apply the concept of potential difference

(voltage) to explain the action of a battery in a circuit.

• To understand how potential difference (voltage) is

distributed in different parts of series and parallel circuits.

• To understand the quantitative relationship between

potential difference and current for a resistor (Ohm’s law).

• To examine Kirchhoff’s circuit rules.

OVERVIEW

In a previous lab you explored currents at different points in

series and parallel circuits. You saw that in a series circuit, the

current is the same through all elements. You also saw that in

a parallel circuit, the sum of the currents entering a junction

equals the sum of the currents leaving the junction.

You have also observed that when two or more parallel

branches are connected directly across a battery, making a

change in one branch does not affect the current in the other

branch(es), while changing one part of a series circuit changes

the current in all parts of that series circuit.

In carrying out these observations of series and parallel

circuits, you have seen that connecting light bulbs in series

results in a larger resistance to current and therefore a smaller

current, while a parallel connection results in a smaller

resistance and larger current.

L04-2 Lab 4 - Ohm’s Law & Kirchhoff's Circuit Rules


In this lab, you will first examine the role of the battery in

causing a current in a circuit. You will then compare the

potential differences (voltages) across different parts of series

and parallel circuits.

Based on your previous observations, you probably associate a

larger resistance connected to a battery with a smaller current,

and a smaller resistance with a larger current. You will explore

the quantitative relationship between the current through a

resistor and the potential difference (voltage) across the

resistor. This relationship is known as Ohm's law. You will

then use Kirchhoff's circuit rules to completely solve a DC

circuit.

INVESTIGATION 1: BATTERIES AND VOLTAGES IN SERIES CIRCUITS

So far you have developed a current model and the concept of

resistance to explain the relative brightness of bulbs in simple

circuits. Your model says that when a battery is connected to a

complete circuit, there is a current. For a given battery, the

magnitude of the current depends on the total resistance of the

circuit. In this investigation you will explore batteries and the

potential differences (voltages) between various points in

circuits.

In order to do this you will need the following items:

• three voltage probes

• two 1.5 volt D batteries (must be very fresh, alkaline)

and holders

• six wires with alligator clip leads

• two #14 bulbs in sockets

• momentary contact switch

You have already seen what happens to the brightness of the

bulb in circuit 1-1 (a) if you add a second bulb in series as

shown in circuit 1-1 (b). The two bulbs are not as bright as the

original bulb. We concluded that the resistance of the circuit is

larger, resulting in less current through the bulbs.

A B

C

Figure 1-1

Lab 4 - Ohm’s Law & Kirchhoff's Circuit Rules L04-3


Figure 1-1 shows series circuits with (a) one battery and one

bulb, (b) one battery and two bulbs and (c) two batteries and

two bulbs. (All batteries and all bulbs are identical.)

Prediction 1-1: What do you predict would happen to the

brightness of the bulbs if you connected a second battery in

series with the first at the same time you added the second

bulb, as in Figure 1-1 (c)? How would the brightness of bulb A

in circuit 1-1(a) compare to bulb B in circuit 1-1 (c)? To

bulb C?

Activity 1-1: Battery Action

1. Connect the circuit in Figure 1-1 (a). Record your

observations about the brightness of the bulb.

2. Now connect the circuit in Figure 1-1(c). [Be sure that the

batteries are connected in series – the positive terminal of

one must be connected to the negative terminal of the

other.] Record your observations about the brightness of

the bulb.



Question 1-1: What do you conclude about the current in the

two bulb, two battery circuit as compared to the single bulb,

single battery circuit?

Prediction 1-2: What do you predict about the brightness of

bulb D in Figure 1-2 compared to bulb A in Figure 1-1 (a)?

D

Figure 1-2

3. Connect the circuit in Figure 1-2 (a series circuit with two

batteries and one bulb). Only close the switch for a moment

to observe the brightness of the bulb – otherwise, you will

burn out the bulb.

Question 1-2: How does increasing the number of batteries

connected in series affect the current in a series circuit?

When a battery is fresh, the voltage marked on it is actually a

measure of the emf (electromotive force) or electric potential

difference between its terminals. Voltage is an informal term

for emf or potential difference. We will use these three terms

interchangeably.



Let's explore the potential differences of batteries and bulbs in

series and parallel circuits to see if we can come up with rules

for them as we did earlier for currents.

How do the potential differences of batteries add when the

batteries are connected in series or parallel? Figure 1-3 shows

a single battery, two batteries identical to it connected in series,

and two batteries identical to it connected in parallel.

Figure 1-3

Prediction 1-3: If the potential difference between points 1

and 2 in Figure 1-3 (a) is known, predict the potential

difference between points 1 and 2 in 1-3 (b) (series connection)

and in 1-3 (c) (parallel connection). Explain your reasoning.

Activity 1-2: Batteries in Series and Parallel

You can measure potential differences with voltage probes

connected as shown in Figure 1-4.

+

-

+

- VPB VPA

(a)

VPA

+ -

+ - VPB

(b) (c)

VPB

+

- B

A

B A B

VPA

+

- A

+ +

+ + +

+

- - -

-

-

-

Figure 1-4

1. Open the experiment file L04A1-2 Batteries.

2. Connect voltage probe VPA across a single battery (as in

Figure 1-4(a)), and voltage probe VPB across the other

identical battery.



3. Record the voltage measured for each battery below:

Voltage of battery A:______

Voltage of battery B: ______

4. Now connect the batteries in series as in Figure 1-4(b), and

connect probe VPA to measure the potential difference

across battery A and probe VPB to measure the potential

difference across the series combination of the two

batteries. Record your measured values below.

Voltage of battery A:_____

Voltage of A and B in series:_____

Question 1-3: Do your measured values agree with your

predictions? Discuss.

5. Now connect the batteries in parallel as in Figure 1-4(c),

and connect probe VPA to measure the potential difference

across battery A and probe VPB to measure the potential

difference across the parallel combination of the two

batteries. Record your measured values below.

Voltage of battery A:______

Voltage of A and B in parallel:______

Question 1-4: Do your measured values agree with your

predictions? Explain any differences.



Question 1-5: Make up a rule for finding the combined

voltage of a number of batteries connected in series.

Question 1-6: Make up a rule for finding the combined

voltage of a number of identical batteries connected in parallel.

You can now explore the potential difference across different

parts of a simple series circuit. Consider the circuit shown in

Figure 1-5.

Figure 1-5

IMPORTANT NOTE: The switch (S1) should remain open

except when you are making a measurement. It is in the

circuit to save the battery. Use the momentary contact switch

for S1.

-

S1

+

VPB

-

+

VPA VPC

+

-

A

B

Do NOT do it, but what would happen if you wired two batteries of

unequal voltage in parallel, hook any two batteries together “anti-

parallel”, or simply short circuit” a single battery? To a very good

approximation, a real battery behaves as if it were an ideal battery in

series with a resistor. Since this “internal resistance” is usually quite

small, the voltages can cause a tremendous amount of current to flow

which, in turn, will cause the batteries to overheat (and possibly

rupture).



Prediction 1-4: If bulbs A and B are identical, predict how the

potential difference (voltage) across bulb A in Figure 1-5 will

compare to the potential difference across the battery. How

about bulb B?

Activity 1-3: Voltages in Series Circuits

1. Continue to use the experiment file L04A1-2 Batteries.

2. Connect the voltage probes as in Figure 1-5 to measure the

potential difference across bulb A and across bulb B.

Record your measurements below.

Potential difference across bulb A:______

Potential difference across bulb B:______

Potential difference across battery:______

Question 1-7: Formulate a rule for how potential differences

across individual bulbs in a series connection combine to give

the total potential difference across the series combination of

the bulbs. How is this related to the potential difference of the

battery?

INVESTIGATION 2: VOLTAGES IN PARALLEL CIRCUITS

Now you will explore the potential differences across different

parts of a simple parallel circuit.

You will need the following material:


• 1.5 V D cell battery (must be very fresh, alkaline) with

holder

• eight alligator clip leads



• two #14 bulbs with holders

• knife switch


Activity 2-1: Voltages in a Parallel Circuit

1. The experiment file L04A1-2 Batteries should still be open

showing two voltage graphs as a function of time. Clear

any old data.

2. Connect the circuit shown in Figure 2-1. Remember to use

the momentary contact switch for S1 and to leave it open

when you are not taking data.

Figure 2-1

3. Begin graphing, and then close and open the switch S2 a

couple of times.

4. Print out and label one set of graphs for your group.

5. Record your measurements using the Digit Display.

Switch S2 open

Voltage across bulb A:______

Voltage across bulb B:______

Voltage across battery:______

Switch S2 closed

Voltage across bulb A:______

Voltage across bulb B:______

Voltage across battery:______

S1

+

VPB -

+

VPA VPC

+

-

A B

S2

-



Question 2-1: Did closing and opening switch S2 significantly

affect the voltage across bulb A (by more than 20% or so)?

Question 2-2: Did closing and opening switch S2 significantly

affect the voltage across bulb B (by more than 20%)?

Question 2-3: Based on your observations, formulate a rule

for the potential differences across the different branches of a

parallel circuit. How are these related to the voltage across the

battery?

You have now observed that the voltage across a (new) real

battery doesn't change much no matter what is connected to it

(i.e., no matter how much current flows in the circuit). An

ideal battery would be one whose voltage did not change at all,

no matter how much current flowing through it. No battery is

truly ideal (this is especially true for a less than fresh battery),

so the voltage usually drops somewhat when there is

significant current flow.



INVESTIGATION 3: MEASURING CURRENT, VOLTAGE AND RESISTANCE

OFF ON

V

A

COM A 20A V, Ω

Ω

0.245 AC DC A

V

Ω

(a) (b)

Figure 3-1

Figure 3-1a shows a multimeter with voltage, current and

resistance modes, and Figure 3-1b shows the symbols that will

be used to indicate these functions.

The multimeters available to you can be used to measure

current, voltage or resistance. All you need to do is choose the

correct dial setting, connect the wire leads to the correct

terminals on the meter and connect the meter correctly in the

circuit. Figure 3-1 shows a simplified diagram of a multimeter.

We will be using the multimeter to make DC (direct current)

measurements, so make sure the multimeter is set to DC mode.

A current probe or a multimeter used to measure current (an

ammeter) are both connected in a circuit in the same way.

Likewise, a voltage probe or a multimeter used to measure

voltage (a voltmeter) are both connected in a circuit in the

same way. The next two activities will remind you how to

connect them. The activities will also show you that when

meters are connected correctly, they don’t interfere with the

currents or voltages being measured.

You will need:

• digital multimeter

• 1.5 V D battery (must be very fresh alkaline) with holder

• six alligator clip leads

• two #14 bulbs and sockets

• 22 Ω and 75 Ω resistors



Activity 3-1: Measuring Current with a Multimeter

Figure 3-2

1. Set up the basic circuit in Figure 3-2, but without the

ammeter (connect the bulb directly to the battery). Observe

the brightness of the bulb.

2. Set the multimeter to measure current.

Important: Use the 20-amp setting and connect the leads to

the 20-amp terminals on the multimeter.

3. When the multimeter is ready, connect it to the circuit as

shown in Figure 3-2.

Was the brightness of the bulb significantly different than it

was without the ammeter?

What current do you measure? _________

Question 3-1: When used correctly as an ammeter, the

multimeter should measure the current through the bulb

without significantly affecting that current. Does this ammeter

appear to behave as if it is a large or small resistor? Explain

based on your observations. What would be the resistance of a

perfect ammeter? Justify your answer.

A



Activity 3-2: Measuring Voltage with a Multimeter

V

S

Figure 3-3

1. Set up the basic circuit in Figure 3-3, but without the

voltmeter. Observe the brightness of the bulb.

2. Set the multimeter to measure voltage.

Important: Use the volts setting and connect the leads to the

voltage terminals on the multimeter.

3. When the multimeter is ready, connect it to the circuit as

shown in Figure 3-3.

Was the brightness of the bulb significantly different than it

was without the voltmeter?

What voltage do you measure? ___________

Question 3-2: When used correctly as a voltmeter, the

multimeter should measure the voltage across the bulb without

significantly affecting that voltage. Does this voltmeter appear

to behave as if it is a large or small resistor? Explain based on

your observations. What would be the resistance of an ideal

voltmeter?

Activity 3-3: Measuring Resistance with a Multimeter

Next we will investigate how you measure resistance with a

multimeter. In earlier labs, you may have observed that light



bulbs exhibit resistance that increases with the current through

the bulb (i.e. with the temperature of the filament). To make

the design and analysis of circuits as simple as possible, it is

desirable to have circuit elements with resistances that do not

change. For that reason, resistors are used in electric circuits.

The resistance of a well-designed resistor doesn't vary with the

amount of current passing through it (or with the temperature),

and they are inexpensive to manufacture.

It can be shown by application of Ohm’s Law and Kirchoff’s

Circuit Rules (which we’ll get to shortly), that the equivalent

resistance of a series circuit of two resistors (R1 and R2) of

resistance R1 and R2 is given by:

1 2seriesR R R= +

Similarly, the equivalent resistance of a parallel circuit of two

resistors of resistance R1 and R2 is given by:

1 2

1 1 1

parallelR R R

= +

One type of resistor is a carbon composition resistor, and uses graphite suspended in a hard glue binder. It is usually

surrounded by a plastic case with a color code painted on it.

Cutaway view of a carbon

composition resistor

showing the cross sectional area of the graphite

material

Figure 3-4

The color code on the resistor tells you the value of the

resistance and the tolerance (guaranteed accuracy) of this value.

The first two stripes indicate the two digits of the resistance value. The third stripe indicates the power-of-ten multiplier.

The following key shows the corresponding values:

black = 0 yellow = 4 grey = 8

brown = 1 green = 5 white = 9 red = 2 blue = 6

orange = 3 violet = 7



The fourth stripe tells the tolerance according to the following key:

As an example, look at the resistor in Figure 3-5. Its two digits

are 1 and 2 and the multiplier is 103, so its value is 12 x 103, or 12,000 Ω. The tolerance is ± 20%, so the value might actually

be as large as 14,400 Ω or as small as 9,600 Ω.

Brown

Red

Orange

None

Figure 3-5

The connection of the multimeter to measure resistance is

shown in Figure 3-6. When the multimeter is in its ohmmeter

mode, it connects a known voltage across the resistor, and

measures the current through the resistor. Then resistance is

calculated by the meter from Ohm’s law.

Note: Resistors must be isolated by disconnecting them from

the circuit before measuring their resistances. This also

prevents damage to the multimeter that may occur if a voltage

is connected across its leads while it is in the resistance mode.

Figure 3-6

red or none = ± 20% gold = ± 5%

silver = ± 10% brown = ± 1%



1. Choose two different resistors (call them R1 and R2) and

read their codes. Work out the resistances and tolerances.

Show your work.

R1 color code:

R1: ________ Ω ± ________ %

R2 color code:

R2: ________ Ω ± ________ %

2. Set up the multimeter as an ohmmeter and measure the

resistors:

R1: ________ Ω

R2: ________ Ω

Question 3-3: Comment on the agreement.

Prediction 3-1: Calculate the equivalent series resistance of

R1 and R2. Show your work.

3. Measure the resistance of the two resistors in series. Use

alligator clip wires to connect the resistors.

Rseries: ________ Ω



Question 3-4: Discuss the agreement between your prediction

and your measured series resistance.

Prediction 3-2: Calculate the equivalent parallel resistance of

R1 and R2. Show your work.

4. Measure the equivalent resistance of the two resistors in

parallel.

Rparallel: ________ Ω

Question 3-5: Discuss the agreement between your prediction

and your measured parallel resistance.

INVESTIGATION 4: OHM’S LAW

What is the relationship between current and potential

difference? You have already seen that there is only a potential

difference across a bulb or resistor when there is a current

through the circuit element. The next question is how does the

potential difference depend on the current? In order to explore

this, you will need the following:

• current and voltage probes

• variable DC power supply

• ten alligator clip leads



• 10 Ω and 22 Ω resistors

• #14 bulb in a socket

Examine the circuit shown below. A variable DC power

supply is like a variable battery. When you turn the dial, you

change the voltage (potential difference) between its terminals.

Therefore, this circuit allows you to measure the current

through the resistor when different voltages are across it.

VPB CPA DC

Power

Supply

+

-

+ -

+

-

Figure 4-1

Prediction 4-1: What will happen to the current through the

resistor as you turn the dial on the power supply and increase

the applied voltage from zero? What about the voltage across

the resistor?

Ohm’s Law: The voltage, V, across an ideal resistor of

resistance R with a current I flowing through it is given by

Ohm’s Law:

V IR=

Activity 4-1: Current and Potential Difference for a

Resistor

1. Open the experiment file L04A4-1 Ohm’s Law.

2. Connect the circuit in Figure 4-1. Use a resistor of 10 Ω.

Note that the current probe is connected to measure the

current through the resistor, and the voltage probe is

connected to measure the potential difference across the

resistor.



Your instructor will show you how to operate the power

supply.

Warning: Do not exceed 3 volts!

3. Begin graphing current and voltage with the power supply

set to zero voltage, and graph as you turn the dial and

increase the voltage slowly to about 3 volts.

Question 4-1: What happened to the current in the circuit and

the voltage across the resistor as the power supply voltage was

increased? Discuss the agreement between your observations

and your predictions.

4. If it’s not visible already, bring up the display for current

CPA versus voltage VPB. Notice that voltage is graphed on

the horizontal axis, since it is the independent variable in

our experiment.

5. Use the fit routine to verify that the relationship between

voltage and current for a resistor is a proportional one.

Record the slope.

Slope = ______________

Question 4-2: Calculate R from the slope. Show your work.

Calculated R = ________________

6. Now remove the resistor from the circuit and measure R

directly with a multimeter.

Measured R = _______________



Question 4-3: Comment on the agreement between the

calculated and measured values for R.

5. On the same graph, repeat steps 2 and 3 for: (a) a 22 Ω

resistor and (b) a light bulb. Be sure to increase the voltage

very slowly for the light bulb, especially in the beginning.

There should now be three sets of data on the I vs. V graph.

6. Print out one set of graphs for your group.

Question 4-4: Discuss the most significant differences

between the curves for the two resistors.

Question 4-5: Based on your data for the light bulb, does it

obey Ohm’s Law? Explain.



Question 4-6: Based on your data for the light bulb, does it

have a larger “resistance” for low current (cooler bulb) or high

current (hotter bulb)? Use Data Studio to find the “resistance”

of the bulb at a point on the curve where the current is low and

one where it is high. State your assumptions about what is

meant by “the resistance” and show your work.

INVESTIGATION 5: KIRCHHOFF’S CIRCUIT RULES

Suppose you want to calculate the currents in various branches

of a circuit that has many components wired together in a

complex array. The rules for combining resistors are very

convenient in circuits made up only of resistors that are

connected in series or parallel. But, while it may be possible in

some cases to simplify parts of a circuit with the series and

parallel rules, complete simplification to an equivalent

resistance is often impossible, especially when components

other than resistors are included. The application of

Kirchhoff’s Circuit Rules can help you to understand the most

complex circuits.

Before summarizing these rules, we need to define the terms

junction and branch. Figure 5-1 illustrates the definitions of

these two terms for an arbitrary circuit.

A junction in a circuit is a place where two or more circuit

elements are connected together.

A branch is a portion of the circuit in which the current is the

same through every circuit element. [That is, the circuit

elements within the branch are all connected in series with each

other.]



Junction 1

Junction 2

4 Ω

12 V 6 Ω

4 V

6 V

R

Branch 1 Branch 2

Branch 3 4 Ω

6 Ω 12 V

4 V

R

(a) (b)

6 V +

–

+

–

+

– +

–

+

–

+

–

Figure 5-1

Kirchhoff’s Rules

1. Junction Rule (based on charge conservation): The sum of

all the currents entering any junction of the circuit must

equal the sum of the currents leaving.

2. Loop Rule (based on energy conservation): Around any

closed loop in a circuit, the sum of all changes in potential

(emfs and potential drops across resistors and other circuit

elements) must equal zero.

You have probably already learned how to apply Kirchhoff’s

rules in class, but if not, here is a quick summary:

1. Assign a current symbol to each branch of the circuit, and

label the current in each branch (I1, I2, I3, etc.).

2. Assign a direction to each current. The direction chosen

for the current in each branch is arbitrary. If you chose

the right direction, the current will come out positive. If

you chose the wrong direction, the current will eventually

come out negative, indicating that you originally chose the

wrong direction. Remember that the current is the same

everywhere in a branch.

3. Apply the Loop Rule to each of the loops.

(a) Let the voltage drop across each resistor be the product of

the resistance and the net current through the resistor

(Ohm’s Law). Remember to make the sign negative if you

are traversing a resistor in the direction of the current and

positive if you are traversing the resistor in the direction

opposite to that of the current.

(b) Assign a positive potential difference when the loop

traverses from the “–” to the “+” terminal of a battery. If

you are going across a battery in the opposite direction,

assign a negative potential difference.

4. Find each of the junctions and apply the Junction Rule to it.



+

–

+

– Loop 1 Loop 2

I1

I1

R3

I2

I2

R2 R1 Junction 2

Junction 1

ε1 ε2

Arbitrarily assigned loop

direction for keeping

track of currents and

potential differences.

Current direction

through battery often

chosen as in direction

of – to +

I3

Figure 5-2.

Now we’ll look at an example. In Figure 5-2 the directions for

the loops through the circuits and for the three currents are

assigned arbitrarily. If we assume that the internal resistances

of the batteries are negligible (i.e. that the batteries are ideal),

then by applying the Loop Rule we find that

Loop 1 1 3 3 1 1 0I R I Rε − − = (1)

Loop 2 2 3 3 2 2 0I R I Rε − − = (2)

By applying the Junction Rule to junction 1 (or 2), we find that

1 2 3I I I+ = (3)

It may trouble you that the current directions and directions

that the loops are traversed have been chosen arbitrarily. You

can explore this issue by changing these choices, and analyzing

the circuit again. You’ll find (assuming no algebraic errors, of

course) that you get the same answers.

Pre-Lab Assignment: Solve Equations 1 through 3 for the

currents 1I , 2I and 3I in terms of the resistances 1R , 2R and

3R and the emf’s 1ε and 2ε . Write your resulting equations:



Activity 5-1: Testing Kirchhoff’s Rules with a Real Circuit

In order to do the following activity you'll need a couple of

resistors and a multimeter as follows:

• two resistors (rated values of 39 Ω and 75 Ω, both

+ 5%)


• 6 V battery

• 1.5 V D battery (very fresh, alkaline) and holder

• 200 Ω potentiometer (to be set to 100 Ω)

• eight alligator clip lead wires

1. Measure the actual values of the two fixed resistors and the

two battery voltages with your multimeter. Record the

results below.

Measured voltage (emf) of the 6 V battery 1ε :_______

Measured voltage (emf) of the 1.5 V battery 2ε :_______

Measured resistance of the 75 Ω resistor 1R :_______

Measured resistance of the 39 Ω resistor 2R :_______

Figure 5-3

A potentiometer (shown in Figure 5-3) is a variable resistor. It

is a strip of resistive material with leads at each end and

another lead connected to a “wiper” (moved by a dial) that

makes contact with the strip. As the dial is rotated, the amount

of resistive material between terminals 1 and 2, and between 2

and 3, changes.

2. Using the resistance mode of the multimeter measure the

resistance between the center lead on the variable resistor

and one of the other leads.



Question 5-1: What happens to the resistance reading as you

rotate the dial on the variable resistor clockwise?

Counterclockwise?

3. Set the variable resistor so that there is 100 Ω between the

center lead and one of the other leads.

4. Wire up the circuit pictured in Figure 5-2 using the variable

resistor set at 100 Ω as 3R . Spread the wires and circuit

elements out on the table so that the circuit looks as much

like Figure 5-2 as possible. [It will be a big mess!]

Note: The most accurate and easiest way to measure the

currents with the digital multimeter is to measure the voltage

across a resistor of known value, and then use Ohm’s Law to

calculate I from the measured V and R.

Pay careful attention to the “+” and “-” connections of the

voltmeter, so that you are checking not only the magnitude of

the current, but also its direction.

5. Use the multimeter to measure the voltage drops across the

resistors and enter your data in Table 5-1 (don’t forget to

use appropriate units!). Fill in the rest of the table:

Calculate the corresponding currents and the percent

difference between these values and those of the pre-lab.

Table 5-1 Results from test of Kirchhoff's Circuit Rules

nominal

R nominal

I measured

R measured

V calculated

I % Difference

R1

R2

R3



Question 5-2: Discuss how well your measured currents agree

with the pre-lab values and consider possible sources of

uncertainty. Were the directions of the currents confirmed?

Question 5-3: What characteristic(s) of real batteries would

lead us to expect that your experimentally determined currents

would be less than predicted? Discuss.

L05-1


Name _____________________________ Date ___________ Partners____________________________

Lab 5 – CAPACITORS & RC CIRCUITS

OBJECTIVES

• To define capacitance and to learn to measure it with a

digital multimeter.

• To explore how the capacitance of conducting parallel

plates is related to the area of the plates and the separation

between them.

• To explore the effect of connecting a capacitor in a circuit

in series with a resistor or bulb and a voltage source.

• To explore how the charge on a capacitor and the current

through it change with time in a circuit containing a

capacitor, a resistor and a voltage source.

OVERVIEW

Capacitors are widely used in electronic circuits where it is

important to store charge and/or energy or to trigger a timed

electrical event. For example, circuits with capacitors are

designed to do such diverse things as setting the flashing rate

of Christmas lights, selecting what station a radio picks up, and

storing electrical energy to run an electronic flash unit. Any

pair of conductors that can be charged electrically so that one

conductor has positive charge and the other conductor has an

equal amount of negative charge on it is called a capacitor.

A capacitor can be made up of two arbitrarily shaped blobs of

metal or it can have any number of regular symmetric shapes

such as one hollow metal sphere inside another, or a metal rod

inside a hollow metal cylinder.

+

+ + + + +

- -

-

- -

-

+

equal and opp charges

-

-

-

+ +

+

-

- -

- -

-

+

Arbitrarily Shaped

Capacitor Plates Cylindrical

Capacitor Parallel Plate

Capacitor

+

+ +

+ +

-

-

-

- +

-

- - -

Figure 1-1: Some different capacitor geometries

L05-2 Lab 5 - Capacitors and RC Circuits


The type of capacitor that is the easiest to analyze is the

parallel plate capacitor. We will focus exclusively on these.

Although many of the most interesting properties of capacitors

come in the operation of AC (alternating current) circuits

(where current first moves in one direction and then in the

other), we will limit our present study to the behavior of

capacitors in DC (direct current) circuits.

The circuit symbol for a capacitor is a simple pair of lines as

shown in Figure 1-2. Note that it is similar to the symbol for a

battery, except that both parallel lines are the same length for

the capacitor.

Figure 1-2: The circuit diagram symbol for a capacitor

In Investigation 1 we will measure the dependence of

capacitance on area and separation distance. In Investigation 2

we shall learn how capacitances react when charge builds up

on their two surfaces. We will investigate what happens to this

charge when the voltage source is removed and taken out of the

circuit.

INVESTIGATION 1: CAPACITANCE, AREA AND SEPARATION

The usual method for transferring equal and opposite charges

to the plates of a capacitor is to use a battery or power supply

to produce a potential difference between the two conductors.

Electrons will then flow from one conductor (leaving a net

positive charge) and to the other (making its net charge

negative) until the potential difference produced between the

two conductors is equal to that of the battery. (See Figure 1-3.)

In general, the amount of charge needed to produce a potential

difference equal to that of the battery will depend on the size,

shape, location of the conductors relative to each other, and the

properties of the material between the conductors. The

capacitance of a given capacitor is defined as the ratio of the

magnitude of the charge, q (on either one of the conductors) to

the voltage (potential difference), V, applied across the two

conductor:

/C q V≡ (1)

Lab 5 -Capacitors and RC Circuits L05-3


A

V

d = separation A = area V = voltage

d

+

-

+q -q

-

-

-

-

-

-

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

- -

Figure 1-3: A parallel plate capacitor with a voltage V across it.

Activity 1-1: Predicting the Dependence of Capacitance on

Area and Separation.

Consider two identical metal plates of area A that are separated

by a distance d. The space between the plates is filled with a

non-conducting material (air, for instance). Suppose each plate

is connected to one of the terminals of a battery.

Prediction 1-1: Suppose you now double the area of each

plate. Does the voltage between the plates change (recall that

the plates are still connected to the battery)? Does the amount

of charge on each plate change? Since /C q V= , how must the

capacitance change?



Prediction 1-2: Now return to the original capacitor. The

easiest way to reason the dependence of capacitance on

separation distance is to charge the plates first and then

disconnect the battery. After we do that, the separation

distance is doubled. Can the charge on the plates change?

Does the electric field between the plates change (assume ideal

conditions: plates large compared to separation distance)?

How does the voltage between the plates change? Since

/C q V= , how must the capacitance change?

The unit of capacitance is the farad, F, named after Michael

Faraday. One farad is equal to one coulomb/volt. As you

should be able to demonstrate to yourself shortly, the farad is a

very large capacitance. Thus, actual capacitances are often

expressed in smaller units with alternate notation as shown

below:

micro farad: 1 µF = 10-6

F

nano farad: 1 nF = 10-9

F

pico farad: 1 pF = 10-12

F

[Note that m, µ, and U when written on a capacitor all stand for

a multiplier of 10-6

.]

There are several types of capacitors typically used in

electronic circuits including disk capacitors, foil capacitors,

electrolytic capacitors and so on. You should examine some

typical capacitors. There should be a collection of such old

capacitors at the front of the room.

To complete the next few activities you will need to construct a

parallel plate capacitor and use a multimeter to measure

capacitance.

You'll need the following items:

• A “fat” UVa directory

• two sheets of aluminum foil about the size of a

directory page

• one or several massive objects (e.g., catalogs)

• digital multimeter with a capacitance mode and clip leads



• ruler with a centimeter scale

• digital calipers

You can construct a parallel plate capacitor out of two

rectangular sheets of aluminum foil separated by pieces of

paper. Pages in the UVa directory work quite well as the

separator for the foil sheets. You can slip the two foil sheets on

either side of paper sheets, and weigh the book down with

something heavy like some textbooks. The digital multimeter

can be used to measure the capacitance of your capacitor.

Activity 1-2: Measuring How Capacitance Depends on

Area or on Separation

Be sure that you understand how to use the multimeter to

measure capacitance and how to connect a capacitor to it. If

you are sitting at an even numbered table, then you will

devise a way to measure how the capacitance depends on the

foil area. If sitting at an odd numbered table then you will

devise a way to measure how the capacitance depends on the

separation between foils. Of course, you must keep the other

variable (separation or area) constant.

When you measure the capacitance of your “parallel plates”, be

sure that the aluminum foil pieces are pressed together as

uniformly as possible (mash them hard!) and that they don't

make electrical contact with each other. We suggest you cut

the aluminum foil so it does not stick out past the pages except

where you make the connections as shown in Figure 1-4.

Notice the connection tabs are offset.

Figure 1-4 Shapes of aluminum foil for capacitors.

Hint: To accurately determine the separation distance, simply

count the number of sheets and multiply by the nominal

thickness of a single sheet. To determine the nominal sheet

thickness, use the caliper to measure the thickness of 100 or

more “mashed” sheets and divide by the number of sheets.

3 cm

3 cm

27 cm

20 cm

3 cm



[Note: If you use the page numbers to help with the counting,

don’t forget that there are two numbered pages per sheet!]

If you are keeping the separation constant, a good separation

to use is about five sheets. The area may be varied by using

different size sheets of aluminum foil. Alternatively, simply

slide one sheet out of the book. Make sure you accurately

estimate the area of overlap.

If you are keeping the area constant, use a fairly large area –

almost as large as the telephone/directory book you are given.

A good range of sheets to use for the separation is one to

twenty.

Important: When you measure C with the multimeter, be sure

to subtract the capacitance of the leads (the reading just before

you clip the leads onto the aluminum sheets).

1. Take five data points in either case. Record your data in

Table 1-1.

Table 1-1

Separation

Length

(mm)

Width

(mm)

Area (m2)

Capacitance (nF)

Number

of Sheets Thickness

(µm)

Cleads Cfoil Cfinal

2. After you have collected all of your data, open the

experiment file L05A1-2 Dependence of C. Enter your

data for capacitance and either separation or area from

Table 1-1 into the table in the software. Be sure there is no

“zero” entry in the case of C vs. separation distance. Graph

capacitance vs. either separation or area.

3. If your graph looks like a straight line, use the fit routine

in the software to find its equation. If not, you should try

other functional relationships until you find the best fit.




Question 1-1: What is the function that best describes the

relationship between separation and capacitance or between

area and capacitance? How do your results compare with your

prediction based on physical reasoning?

Question 1-2: What difficulties did you encounter in making

accurate measurements?

The actual mathematical expression for the capacitance of a

parallel plate capacitor of plate area A and plate separation d is

derived in your textbook. The result is

0

AC

dκε= (2)

where

0 8.85pF mε =

and κ is the dimensionless dielectric constant.

Question 1-3: Do your predictions and/or observations on the variation of capacitance with plate area and separation seem to

agree qualitatively with this result?



Question 1-4: Use one of your actual areas and separations to

calculate a value of κ using this equation. Show your

calculations. What value of the dielectric constant of paper do you determine? (The actual dielectric constant varies

considerably depending on what is in the paper and how it was processed.) Typical values range from 1.5 to 6.

κ ________________________

INVESTIGATION 2: CHARGE BUILDUP AND DECAY IN CAPACITORS

Capacitors can be connected with other circuit elements. When they are connected in circuits with resistors, some interesting

things happen. In this investigation you will explore what happens to the voltage across a capacitor when it is placed in

series with a resistor in a direct current circuit.

You will need:

• one current and one voltage probe

• 6 V battery

• #133 flashlight bulb and socket (on RLC board)

• electrolytic capacitor (~23,000 µF)

• six alligator clip wires

• single pole, double throw switch

• RLC circuit board

You can first use a bulb in series with an “ultra capacitor” with

very large capacitance (> 0.02 F!). These will allow you to see what happens. Then later on, to obtain more quantitative

results, the bulb will be replaced by a resistor.

Activity 2-1: Observations with a Capacitor, Battery and

Bulb

1. Set up the circuit shown in Figure 2-1 using the 23,000 µF capacitor.

Be sure that the positive and negative terminals of the capacitor are connected correctly! Because of the chemistry of their

dielectric, electrolytic capacitors have a definite polarity. If hooked up backwards, it will behave like a bad capacitor in

parallel with a bad resistor.



Single pole,

double throw

switch 2

1

V

+ –

+ – C

Figure 2-1

Question 2-1: Sketch the complete circuit when the switch is

in position 1 and when it is in position 2. For clarity, don’t draw components or wires that aren’t contributing to the function of the circuit.

position 1 position 2

2. Move the switch to position 2. After several seconds, switch it to position 1, and describe what happens to the

brightness of the bulb.

Question 2-2: Draw a sketch on the axes below of the

approximate brightness of the bulb as a function of time for the above case of moving the switch to position 1 after it has been

in position 2. Let t = 0 be the time when the switch was moved to position 1.

Time [sec]

Brightness

1 2 3 4 5 6 7 0

3. Now move the switch back to position 2. Describe what happens to the bulb. Did the bulb light again without the

battery in the circuit?

Question 2-3: Draw a sketch on the axes below of the

approximate brightness of the bulb as a function of time when



it is placed across a charged capacitor without the battery

present, i.e. when the switch is moved to position 2 after being

in position 1 for several seconds. Let t = 0 be when the switch is moved to position 2.

Time [sec] Brightness

1 2 3 4 5 6 7 0

Question 2-4: Discuss why the bulb behaves in this way. Is

there charge on the capacitor after the switch is in position 1 for a while? What happens to this charge when the switch is

moved back to position 2?

4. Open the experiment file L05A2-1 Capacitor Decay, and display VPB and CPA versus time.

5. Connect the probes in the circuit as in Figure 2-2 to measure the current through the light bulb and the potential

difference across the capacitor.

Single pole,

double throw switch 2

1

V

+ –

+ –

C

VPB

+ –

CPA

Figure 2-2

6. Move the switch to position 2.

7. After ten seconds or so, begin graphing. When the graph lines appear, move the switch to position 1. When the

current and voltage stop changing, move the switch back to position 2.

8. Print out one set of graphs for your group.



9. Indicate on the graphs the times when the switch was moved from position 2 to position 1, and when it was

moved back to position 2 again.

Question 2-5: Does the actual behavior over time observed on

the current graph agree with your sketches in Questions 2-2 and 2-3? How does the brightness of the bulb depend on the

direction and magnitude of the current through it?

Question 2-6: Based on the graph of potential difference

across the capacitor, explain why the bulb lights when the switch is moved from position 1 to position 2 (when the bulb is

connected to the capacitor with no battery in the circuit)?

Activity 2-2: The Rise of Voltage in an RC Circuit

We will now look at a circuit which we can quantitatively

analyze. A light bulb, as we have seen, has a very non-linear relationship between the applied voltage and the current

through it. A resistor, on the other hand, obeys Ohm’s Law: the voltage across a resistor is proportional to the current

through the resistor. Similarly, the voltage across a capacitor is proportion to the charge on the capacitor.

Single pole,

double throw

switch 2

1

V

+ –

+ – C R

Figure 2-3

Consider Figure 2-3 (the same as Figure 2-1, but with the bulb replaced by a resistor). We will assume that the switch has

been in position 2 “for a very long time” so that the capacitor is fully discharged. [We will soon get a sense of how long one

must wait for it to be “very long”.]



We have seen that when the capacitor is fully discharged, there will be no current flowing in the circuit..

V

+ –

+ –

C R

+ –

Figure 2-3

Now we consider what happens when we move the switch to position 1. From Kirchoff’s Junction Rule (charge

conservation), we can see that any current that flows through the capacitor also flows through the resistor and the battery

(they are in series):

capacitor resistor battery

I I I= = (1)

From Kirchoff’s Loop Rule (energy conservation), we see that

the voltage drop across the resistor plus the voltage drop across the capacitor is equal to the voltage rise across the battery:

/resistor capacitor resistor capacitor battery

V V I R q C V+ = + = (2)

As I dq dt= , we have a simple differential equation:

battery

Rdq dt q C V+ = (3)

Integrating this yields:

( )( ) 1 t RC

batteryq t CV e−

= − (4)

Finally, the voltage across the capacitor (which we can

measure) will be given by:

( )( ) 1 t RC

capacitor batteryV t V e−

= − (5)

Now we can see what “a long time” means. As the argument

of the exponential function is unitless, the quantity RC must

have units of time. RC is called the time constant of the

circuit.

Initially, at 0t = , there is no charge on the capacitor and all of

the battery’s voltage will be across the resistor. A current

(equal to V R ) will then flow, charging the capacitor.

As the capacitor charges, the voltage across the capacitor will increase while that across the resistor will decrease. For times

short relative to RC , the charge on the capacitor will increase

essentially linearly1 with respect to time.

1 1 x

e x−

− ≈ , for 1x≪ .



As the capacitor charges, however, the voltage across it will increase, forcing the voltage across the resistor to decrease.

This means that the current will also decrease, which will lead to a drop in the rate of charging. Asymptotically (for times

large relative to RC ), the capacitor’s voltage will approach the

battery’s voltage and there will be no further current flow.

1. Open the experiment file L05A2-2 RC Circuit. This will take data at a much higher rate than before, and will allow

us to graph the charging of the capacitor, using a smaller C which we can readily measure with the multimeter.

2. Replace the light bulb in your circuit (Figure 2-2) with a

100 Ω resistor, and the large capacitor with one in the

80 µF to 120 µF range. [Use the RLC circuit board.]

Move the switch to position 2.

3. Begin graphing and immediately move the switch to

position 1. Data taking will start when the switch is moved and cease automatically.

NOTE: DataStudio is configured in this activity to start taking data when the voltage sensed by VPB starts rising. Make sure

that you have hooked the probes up correctly or it won’t start.

4. You should see an exponential curve which, for the charging of the capacitor, is:

( ) ( )1 t RC

fV t V e−

= − (3)

Use the Smart Tool to determine from your graph the time

constant (the time for the voltage across the capacitor to

reach 63% [actually 1 1 e− ] of its final value – after the

switch is moved to position 1). Record your data below.

[Don’t forget units!]

Final voltage:________

63% of final voltage:________

Time constant:________



5. Convince yourself from the equation above that the

time constant must equal exactly RC. Remove the

components from the circuit and then measure R and

C with the multimeter and calculate RC. [Don’t forget

units!]

R:________

C:________

RC:_______

Question 2-7: Discuss the agreement between the measured

time constant and RC.

Agreement (%): __________________

6. Now you can use the software to fit an “Inverse

Exponent Fit” to your data. Choose the range of times

that you want to fit. The data box should already be

present on your voltage graph. Look at the root mean

square error. It should be much less than one.

Equation of curve fit:

Parameters for a good fit:




Question 2-8: What is the physical significance of the

parameters “A”, “B”, and “C” in your fit?

Question 2-9: Calculate the time constant from your fit, and

compare to what you found from your measured values of R

and C. Discuss the agreement.

Parameter(s): ________________________

Calculation of time constant from parameter(s):

Calculated RC (from step 5): ________________

Agreement (%): __________________



L06-1


Name ________________________ Date__________ Partners

Lab 6 - ELECTRON

CHARGE-TO-MASS RATIO

OBJECTIVES

• To understand how electric and magnetic fields impact an

electron beam

• To experimentally determine the electron charge-to-mass ratio

OVERVIEW

In this experiment, you will measure e/m, the ratio of the electron’s

charge e to its mass m. Given that it is also possible to perform a

measurement of e alone (the Millikan Oil Drop Experiment), it is

possible to obtain the value of the mass of the electron, a very

small quantity.

If a particle carrying an electric charge q moves with a velocity v

in a magnetic field B that is at a right angle to the direction of

motion, it will experience the magnetic part of the Lorentz force:

F = qv x B (1)

Which, because of the vector product, is always perpendicular to

both the magnetic field and the direction of motion. A constant

force that is always perpendicular to the direction of motion will

cause a particle to move in a circle. We will use this fact to

determine e/m of the electron by measuring the radius of that

circle. To this end we must:

• produce a narrow beam of electrons of known energy,

• produce a uniform magnetic field,

• find a way to measure the radius r of the circular orbit of the

electrons in that magnetic field,

and

• find the relation between that radius and the ratio e/m.

We will discuss these tasks in order.

The Electron Beam

When one heats a piece of metal, say a wire, to 1,000 K or beyond,

electrons will “boil off” from its surface. If one surrounds the wire

L06-2 Lab 6 - Electron Charge-to-Mass Ratio


with a positively charged electrode, an anode, the electrons will be

attracted to it and move radially outward as indicated in Figure 1.

On their way to the anode they will acquire a kinetic energy

eVmvEk

==2

21 , (2)

where V is the potential difference, or voltage, between the heated

filament, called the cathode, and the anode.

Figure 1

Most of the electrons will strike the anode. However, if one cuts a

narrow slit into the anode, those electrons that started out toward

the slit will exit through it as a narrow beam with a kinetic energy

Ek.

The Magnetic Field

According to Ampere’s law a wire

carrying a current I is surrounded by

a magnetic field B, as shown in

Figure 2. If the wire is bent into a

circle the field lines from all sides

reinforce each other at the center,

creating an axial field (see Figure 3).

Usually one will not use a single

loop of wire to create a field but a

coil with many turns.

Figure 3 Magnetic field of a wire loop.

Figure 2 Magnetic field of

a straight wire.

Lab 6 - Electron Charge-to-Mass Ratio L06-3


If one uses two coaxial coils of radius R that are a distance d apart,

as shown in Figure 4, the field at the center point between the coils

will be nearly homogeneous1. H. von Helmholtz (1821-1894)

realized that there remains a free parameter, namely the coil

separation d, that can still be adjusted. He showed that when

d = R, the result is a particularly homogeneous field in the central

region between the coils.2 Since that time Helmholtz coils have

been used when there is a need for homogeneous magnetic fields.

Figure 4 Magnetic field B of a pair of Helmholtz coils.

One can show that the field in the center of a Helmholtz coil is

given by

,55

8 0 IR

NB

=

µ (3)

where I is the current flowing through both coils, R is their mean

radius, N is the number of turns of wire in each coil, and

µ0 = 4π × 10-7

T·m/A is the permeability constant.

The Electron Orbit

Experiments like the one that you will perform have been used to

measure the mass of charged particles with great precision. In these

experiments the particles move in a circular arc whose beginning and

end are measured very accurately in a near perfect vacuum. For our

simple experiment we cannot go to such lengths and a simple

expedient has been used to make the electron orbit visible. The bulb

surrounding the electron source is filled with helium vapor. When the

electrons collide with the atoms, the atoms emit light so that one can

follow the path of the electron beam. These collisions will diminish

1 The symmetry of the arrangement makes the first derivative of the field with respect to

the axial direction vanish.

2 The second derivative vanishes as well.



the accuracy of the experiment but it remains adequate for our

purposes.

A particle moving in a circle of radius r must be held there by a

centripetal force

r

mvFc

2

= . (4)

In our case, that centripetal force is provided by the magnetic part

of the Lorentz force, Equation (1), hence

r

mvevB

2

= (5)

This equation contains the velocity v, which we can eliminate by

using Equation (2). Rewriting Equation (2), we find

m

eVv

2= (6)

and hence

2

2 2 2 2 2 2

0

2 125

32

e V R V

m B r N I rµ

= =

(7)

In this equation, V is the voltage between cathode and anode and r

is the mean radius of the circular electron orbit, both of which can

be measured, and B is the magnetic field through which the

electrons pass. We know the magnetic field at the center of the

Helmholtz coil, which can be obtained, using Equation (3), from a

measurement of the current through the coils, the dimension of the

coils and the number of turns. The magnetic field does not change

very much away from the center of the coils.

INVESTIGATION 1: FINDING e/m

For this investigation, you will need the following:

• Helmholtz coil with e/m glass tube

• Bar magnet

• Meter stick

Activity 1-1: The e/m Apparatus

In this activity, you will familiarize yourself with the setup you

will be using.



Figure 5 e/m Apparatus

1. Turn the main power on. The unit will perform a self test

lasting no more than 30 s. Do not do anything with the unit

during the self test. When it is finished, the coil current display

will be stabilized and indicate “000”. The unit is now ready to

use, but note that there is a ten minute warm-up time before

you should take final measurements. You can go ahead and

proceed with the remainder of this procedure.

2. Look in the center of the Helmholtz coils for the glass tube.

The electrons will follow circular orbits inside this evacuated

glass tube. The tube has a tiny bit of helium vapor inside of it.

The energetic electrons collide with and ionize the helium

atoms, causing the gas to glow and making the beam visible.

The glass tube is extremely fragile, so be very careful around it.

3. Locate the grid and anode inside the glass tube. It is the pair of

vertically oriented metal cylinders with a gap between them.

At its top and center is the filament or cathode that will be

heated by a current to emit the electrons.

4. You should now be able to see the filament (wires that are

glowing orange due to being heated).



Figure 6 Grid and Anode

5. There are three separate electrical circuits: 1) to heat the

filament/cathode (over which you have no control); 2) to apply

a voltage between cathode and anode (denoted as Accelerating

Voltage on the unit); 3) to supply the current (denoted as Coil

Current) for the Helmholtz coils.

6. Measure the diameter of the Helmholtz coils in several places

and take the average. Record the mean radius R below.

Radius R: _______________

7. Measure the mean separation d between the coils. You may

want to average several measurements here also.

Coil separation d: __________________

Verify that d R≈ .

8. The manufacturer states that there are 130 turns in each coil.

9. Calculate the “constant of proportionality” between the current

passing through the Helmholtz coils and the magnetic field

produced. You will need the above parameters to do this.

Look at Equation (3).

BHelmoltz (Tesla) = __________ × I (amps)



10. Turn up the voltage adjust knob to a voltage of about 200 V.

Look for the electron beam, which should be pointing down.

NOTE: Both the voltage and current outputs are controlled by a

microprocessor, which locks out the controls at both the minimum

and maximum settings. There is not a manual “stop” on the knobs.

When the knob reaches the maximum setting, it will still turn, but

the appropriate value will not change. This feature prevents

excessive voltage being applied to the tube or excessive current

through the coils.

11. Turn the Current Adjust control up and observe the circular

deflection of the beam. When the current is high enough, the

beam will form a circle. The diameter of the electron path in

the magnetic field can be measured using the etched glass

internal scale in the tube. The graduations and numerals of the

scale are illuminated by the collision of the electrons, making

observation reading fairly easy. Vary the Current Adjust and

note the electron beam striking several of the centimeter scale

markings. You should also be able to see a vertical line

indicating the half-centimeter mark. The scale numbers

fluoresce when the beam hits them.

NOTE: Sometimes the electron beam slightly misses the internal

glass scale tube. DO NOT TRY TO MAKE ANY

ADJUSTMENTS! Sometimes it helps to put a dim backlight

behind the scale to help see the numbers and half-centimeter

marks. [If it is really bad, ask your TA to look at it.]

12. Describe what happens to the beam of electrons as the coil

current is increased:

Question 1-1: What causes this behavior?



13. Set the coil current to 1.7 A. Adjust the accelerating voltage

while looking at the electron beam path..

Question 1-2: What do you observe as the accelerating voltage is

changed while keeping the coil current constant? Explain why this

occurs.

14. While the electron beam is somewhere near the middle of the

glass rod, use the bar magnet to see how it affects the electron

beam.

Question 1-3: Describe what you observe as you move the bar

magnet around. Can you produce a helical path for the electron?

Can you see why you want to keep spurious magnetic fields away

from the electron beam?

Question 1-4: The nominal value for the apparatus you will be

using is 1.3 mT (milliTesla). The Earth’s magnetic field is

approximately 0.1 mT and is pointing into the ground at an angle

of about 58° with respect to the horizontal. Discuss how much

difficulty the Earth’s magnetic field will cause in your experiment.



Question 1-5: If it were possible to arbitrarily orient the

apparatus, in what direction (parallel, anti-parallel, perpendicular,

other) should it be aligned in order to minimize the effects of the

Earth’s magnetic field? Explain your reasoning.

NOTE: We cannot compensate for the Earth’s magnetic field with

this apparatus.

15. Make sure you place the bar magnet at the other end of the

table from your apparatus to minimize any possible effects.

Activity 1-2: Measurement Of Charge-To-Mass Ratio

NOTE: The electrons collide with the gas atoms that were

introduced to make the beam visible. Unavoidably, the electrons

lose some energy in the collisions (that make the beam visible).

In order to minimize this effect, you should concentrate on those

electrons that have the highest energy: those at the outer edge of

the beam (largest radius). Ask your Instructor if you are

uncertain about this.

1. We will make measurements for the electrons moving in

various circular orbits. You will write down the diameter of

the path by reading the illuminated tube. You will need these

in order to determine e/m from Equation (7).

2. Set the anode voltage to 300 V. Change the coil current until

the beam hits the 5 cm mark. This will be the diameter, not

the radius of the orbit.

3. Record the coil current and diameter in Table 1-1. Decrease

the coil current to go through, in turn, each 0.5 cm mark on the

scale. The 0.5 cm marks are a vertical line between the

numbered cm marks. This can proceed quite rapidly by one

person observing the beam while changing the coil current and

another person filling in Table 1-1.

4. Proceed in turn to observe and record data for the other

accelerating voltages in Table 1-1. Sometimes the beam will



physically not be present for every position listed in Table 1-1.

Leave the table blank when this occurs.

5. Turn the main power off.

Table 1-1

6. In Excel, open L06-Table 1-1.xls (found in the same location

as are the DataStudio files). Select tab “Data” and key in your

coil current data from Table 1-1.

7. Select tab “e_m”. Note that the data has now been re-sorted

into one table. In cell D2 (the first empty cell in the e/m

column), enter an Excel version of Equation (7) using the

contents of the corresponding Diameter, Voltage, and Current.

[Don’t forget to change your values to the SI system before

determining your final value of e/m!]

8. Click on the lower right corner of the cell and “drag” the

formula into the rest of the e/m cells.

Accelerating (anode) Voltage

300 V 350 V 400 V 450 V 500 V

Diameter

(cm)

Coil

Current

(A)

Coil

Current

(A)

Coil

Current

(A)

Coil

Current

(A)

Coil

Current

(A)

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0



9. Set up column E to be the difference between your value of e/m

and the accepted one (1.759 x 1011

C/kg).

10. Average all your results in Excel to obtain a mean value for

e/m.

e/m:

11. Find the statistical uncertainty of your average value. [Hint:

Think “standard error of the mean”]

Uncertainty:

12. Compare your value with the accepted value of e/m.

Error: ________%

Question 1-5: Discuss well your result compare with the accepted

value.

13. Print out one data table for your group. You only need one

printout per group.

Activity 1-3: Investigation of Systematic Uncertainty

It is useful in an experiment like this to see if there are systematic

uncertainties that might affect your final results. From

Equation (7) [and a review of Appendix C, of course!] we can see

that the relative uncertainty in our experimental determination of

e/m due to uncertainties in our measured quantities (V, I, R, and r)

is given by:

2 2 2 2 2

2 2 2e m V I R r

e m V I R r

σ σ σ σ σ = + + +

(8)



Question 1-6: With Equation (8) in mind, discuss possible

sources of uncertainty in your experiment.

1. It may be possible that we may learn something about our

uncertainties if we compare the values of e/m versus our

parameters. Because we have our data in Excel, it is quite easy

to do that.

2. Plot all your values of e/m versus a) accelerating voltage, b)

coil current, and c) orbit radius. Produce trendline fits for your

data.

3. Print out one copy each of these graphs and include them in

your report.

Question 1-6: Look carefully at the three plots you just made and

at your trendlines. Do you see any patterns? Could these possibly

indicate any systematic problems? Discuss these possible

uncertainties and how you might be able to correct for them or

improve the experiment. List any other systematic uncertainties

you can think of and discuss.

PLEASE CLEAN UP YOUR LAB AREA AND MAKE SURE

THAT THE POWER IS OFF!

L07-1


Name: _______________________ Date ____________ Partners ______________________________

Lab 7 – INDUCTORS AND LR CIRCUITS

The power which electricity of tension possesses of

causing an opposite electrical state in its vicinity has

been expressed by the general term Induction . . .

Michael Faraday

OBJECTIVES

• To explore the effect of the interaction between a magnetic

field and a coil of wire (an inductor).

• To explore the effect of an inductor in a circuit with a

resistor and voltage source when a constant (DC) signal is

applied.

• To explore the effect of an inductor in a circuit with a

resistor and voltage source when a changing signal is

applied.

OVERVIEW

You have seen that resistors interact with DC signals (currents

or voltages) to produce voltages and currents which can be

predicted using Ohm’s Law:

R

V IR= (1)

You have also seen that the corresponding relationship for

capacitors is

/C

V q C= (2)

where

dq

Idt

= (3)

L07-2 Lab 7 - Introduction to Inductors and L-R Circuits


Capacitors in RC circuits give predictable currents and voltages

according to a different relationship. For the example of a

discharging capacitor in an RC circuit, the voltage across the

capacitor is given by 0

t RC

CV V e

−

= .

In this laboratory you will be introduced to yet another circuit

element, the inductor (typically denoted by an L). An inductor

is basically a coil of wire. A time varying magnetic flux ( )tΦ

in such a coil induces a voltage across the coil according to

L

dV

dt

Φ= − (4)

where

AB d∫ •=Φ

AreaCoil

(5)

On the other hand, a current I flowing through a coil produces

a magnetic flux proportional to I. So, a time varying current in

a coil will generate a “back emf”

L

d d dIV

dt dI dt

Φ Φ= − = − (6)

We defined the inductance (more properly, the self inductance)

as

d

LdI

Φ≡ (7)

Hence, the analog of Ohm’s Law for inductors is

L

dIV L

dt= − (8)

L is a constant whose value is a function of the geometry of the

coil).

Similarly, a second coil exposed to the field of the first will

have a voltage

12

dIV M

dt= − (9)

induced in it. M is called the mutual inductance and is a

constant determined by the geometry of the two coils. Such

coil pairs are called “transformers” and are often used to “step-

up” or “step-down” voltages.

Lab 7 - Introduction to Inductors and L-R Circuits L07-3


INVESTIGATION 1: THE INDUCTOR

The purpose of this investigation is to introduce the behavior of

coils of wire (inductors) in the presence of magnetic fields and

in particular for changing magnetic fields.

You will need the following materials:

• voltage probe and current probe

• small compass

• bar magnet

• large coil of wire (inductor) (approximately 3,400 turns,

800 mH and 63 Ω)

• 2,000-turn detector coil

• 6 volt battery

• alligator clip leads

• switch

Activity 1-1: Magnetic Fields and Inductors, Part I

Magnetic effects are usually described by the existence of a

magnetic field. A magnetic field can exert a force on a

magnetized object, such as a compass needle. In this activity

you will investigate the effect of a magnetic field on an isolated

coil of wire (an inductor). One can verify the presence of a

magnetic field at a point in space by using a simple compass.

Lay your bar magnet on the sheet below as shown. Use a small

compass to determine the direction of B. Make sure

extraneous metal is not affecting the compass. The direction of

the compass needle indicates the direction of the magnetic

field. Indicate with arrows at the ×’s the direction in which the

compass needle points in the vicinity of the bar magnet. Try

enough of the ×'s to draw the magnetic field lines.

N S

x x

x

x

x

x x

x

x

x x x

x

x

x

x x

x

x x

x

x

x

x



One surprising property of magnetic fields is the effect they

can have on wires. It is especially noticeable with a coil of

many turns of wire, since this will magnify the effect. With

your large coil connected to the voltage probe, you will

observe the effects of a magnetic field in the vicinity of the

coil.

N

S

VPA

Figure 1

Prediction 1-1: Consider Figure 1 above. Predict the reading

(steady positive, negative but heading positive, zero, etc.) of

the voltage probe, VPA, when the magnet is

(a) held motionless outside the coil along the axis as shown.

(b) held motionless inside the coil along the axis.

(c) moved quickly from outside the coil to inside the coil, and

then back out.

Now we will test your predictions.

1. Connect the large coil (inductor) to the voltage probe as

shown in Figure 1. Make sure nothing else is connected to

the coil. (For this exercise, the polarity of VPA is

arbitrary.)

2. Open the experiment file called L07A1-1 Measure Coil

Voltage.



3. As illustrated above, hold the bar magnet outside the coil

and begin graphing the voltage across the coil. Hold the

magnet motionless outside the coil for a few seconds. Then

move it fairly rapidly inside the coil. Hold the magnet

motionless inside the coil for a few seconds. Finally, move

it fairly rapidly outside the coil. Then stop graphing.

4. Flip the polarity of the magnet, i.e. turn the bar magnet

around. Begin graphing and repeat the above sequence.

Question 1-1: Summarize your observations. Describe the

effects on the coil of wire when you have external magnetic

fields that are a) steady (non changing) and b) changing. Do

your observations agree with your predictions?

Prediction 1-2: Now consider the case where the bar magnet

is held motionless but the coil is moved toward or away from

the magnet. Predict what will be the reading by the voltage

probe.

5. Choose one of the previous motions of the magnet (N or S

pole pointing towards coil, and either moving magnet in or

out). Clear all data. Begin graphing the voltage across the

coil. Repeat that motion of the magnet. Then, hold the

magnet still and move the coil so that the relative motion

between coil and magnet is the same.



Question 1-2: Describe your observations. Is it the absolute

motion of the magnet, or the relative motion between coil

and magnet that matters?

6. Try to change the magnitude of the observed voltage by

moving the magnet in and out faster and slower. Do it two

or three times on the same display.

7. Print out the results.

Question 1-3: What is the relationship you find between the

magnitude of the voltage and the relative speed between the

magnet and the coil? Explain.

Activity 1-2: Existence of a Magnetic Field Inside a

Current-Carrying Coil.

In the previous activity you used a permanent bar magnet as a

source of magnetic field and investigated the interaction

between the magnetic field and a coil of wire. In this activity

you will discover another source of magnetic field--a current

carrying coil of wire.

Prediction 1-3: Consider the circuit in Figure 2 in which a

coil (an inductor) is connected to a battery. Predict the

direction of the magnetic field at points A (along axis, outside

of the coil), B (along the axis, inside the coil), and C (outside,

along the side of the coil) after the switch is closed. [Hint:

Consider the direction of the current flow.]



switch

V(battery)

× • × ×

×

×

×

×

×

×

×

× × ×

× × ×

× × ×

A B

C × •

•

Figure 2

1. Connect the large coil, switch and 6-volt battery in the

circuit shown in Figure 2.

2. Close the switch.

3. Use the compass to map out the magnetic field and draw

the field lines on the figure. Try enough locations to get a

good idea of the field.

4. Open the switch. Do not touch metal when doing so or

you may receive a small shock. Flip the polarity of the

battery by changing the leads at the battery. Close the

switch again and note the changes to the magnetic field.

Just check a few positions.

5. Open the switch.

Question 1-4: Clearly summarize the results. How do your

observations compare to your observations of the magnetic

field around the permanent magnet? What happened when you

changed the battery polarity (direction of current)?

Summary: In this activity you observed that a current-

carrying coil produces a magnetic field. The magnitude of the

magnetic field is largest in the center of the coil. Along the

axis of the coil the direction of the magnetic field is aligned to

the axis and points consistently in one direction. Outside the

coil, the magnetic field is much weaker and points in a

direction opposite to the magnetic field at the coil axis.



The situation can be pictured as shown in Figure 3 below. On

the left is a coil. On the right is a current-carrying coil and the

resulting magnetic field represented by the vectors B.

I

I

B

B

B

Figure 3

Activity 1-3: Magnetic Fields and Inductors, Part II

You have now observed that a current through a coil of wire

creates a magnetic field inside and around the coil. You have

also observed that a changing magnetic field created by a

moving magnet inside a coil can induce a voltage across the

coil. In this activity you will observe the circumstances under

which interactions between two coils result in an induced

voltage.

Consider the circuit shown in Figure 4 (below), in which the

coil on the left is connected to only the voltage probe, and the

coil on the right is connected to a battery and a contact switch.

VPA

S

V (battery)

Figure 4

Prediction 1-4: Under which of the conditions listed below

will you observe a non-zero voltage across the coil that is

connected to the voltage probe?

Case I: When the switch is closed awhile, and both coils are

held motionless. Circle: yes no



Case II: When the switch is closed awhile, and there is

relative motion between the coils. Circle: yes no

Case III: When the switch is left open awhile. Circle: yes

no

Case IV: At the moment when the switch goes from open to

closed or from closed to open, with both coils motionless.

Circle: yes no

Test your predictions.

1. Connect the circuit in Figure 4 (above). Connect the large

coil to a switch and 6 V battery, and the small detector coil

to a voltage probe.

2. Open the experiment file L07A1-1 Measure Coil Voltage

if it's not already open.

With Data Studio, you may find it easier to set the voltage axis

to a sensitive scale and then prevent automatic re-scaling. To

do this, double-click on the graph, click “Axis Settings”, and

deselect “Adjust axes to fit data”.

3. Describe your observations of the coil voltage below.

Note: when the switch has been closed and then you open

it, you may see a very high frequency, complicated voltage

oscillation that we will learn more about in a later lab. For

now, concentrate on the lower frequency response.

Case I: Switch closed and coils motionless.

Case II: Switch closed, relative motion between coils.

Case III: Switch open.

Case IV: Switch changes position. (Coils must be close

together.)



Question 1-5: Make a general statement about the behavior of

coils (inductors) based on your observations. Include in your

statement the condition(s) under which a voltage is induced in

a coil that is in the vicinity of another coil.

We now want to see what will happen if we replace the battery

and switch in Figure 4 with an AC voltage source.

4. Remove the battery and switch from the large coil, and

instead connect the coil to the output of the PASCO

interface (see Figure 5). A voltage probe (VPA) should still

be connected to the small coil.

VPA

PASCO Interface

Output

Figure 5

5. Open the experiment file L07A1-2 Coil Voltage with AC.

6. With the small coil about a foot away, begin graphing and

slowly move the small coil toward the large coil. When

you're finished, leave the small coil approximately in the

position of maximum signal, to be ready for the next

activity.

Question 1-6: Explain your observations. Comment on the

phase relationship between the voltage driving the large coil,

and the signal detected by the small coil. (Hint: When is the

magnetic field of the large coil changing most rapidly?)



Prediction 1-5: What do think will happen if we leave the

coils motionless, and change the frequency of the AC voltage

driving the large coil? [Assume that the frequencies are such

that the amplitude of the current through the large coil remains

constant.]

Test your prediction.

7. Open the experiment file L07A1-3 Coil Voltage vary Hz.

[To avoid clutter, this will only graph the coil detector

voltage and not the voltage driving the large coil.]

8. Set the frequency to 1 Hz and begin graphing. Repeat with

a frequency of 2 Hz. The two sets of data will be on top of

one another.

Note: We use low frequencies so that the “self-inductance” of

the large coil does not significantly impede the flow of current.

9. Move the detector coil away to prove that the signal is

really from the large coil.

10. Try larger frequencies if you wish, but be aware that the

amplitude of the current in the large coil will not be

constant.

Question 1-7: Describe your observations. Did the detected

voltage change with driving frequency? How did its amplitude

change? Explain why.

Summary: In this investigation you have seen that a changing

magnetic field inside a coil (inductor) results in an induced

voltage across the terminals of the coil.

You saw that such a changing magnetic field can be created in

a number of ways: (1) by moving a magnet in and out of a



stationary coil, (2) by moving a coil back and forth near a

stationary magnet, and (3) by placing a second coil near the

first and turning the current in the coil on and off, either with a

battery and switch or with an AC voltage source.

In the next investigation you will observe the “resistance”

characteristics of an inductor in a circuit.

INVESTIGATION 2: DC BEHAVIOR OF AN INDUCTOR

Physically, an inductor is made from a long wire shaped in a

tight coil of many loops. Conventionally, a symbol like

is used to represent an inductor.

In the simplest case we can model an inductor as a long wire.

In previous investigations we approximated the resistance of

short wires to be zero ohms. We could justify such an

approximation because the resistance of short wires is very

small (negligible) compared to that of other elements in the

circuit, such as resistors. As you may know, the resistance of a

conductor (such as a wire) increases with length. Thus for a

very long wire, the resistance may not be negligible.

All ‘real’ inductors have some resistance which is

related to the length and type of wire used to wind the

coil. Therefore, we model a real inductor as an ideal

inductor (zero resistance) with inductance L in series

with a resistor of resistance RL. A real inductor in a

circuit then can be represented as shown in the

diagram to the right, where the inductor, L, represents an ideal

inductor. For simplicity, usually we let the symbol

represent an ideal inductor while remembering that a real

inductor will have some resistance associated with it.

In this investigation you will need the following materials:

• inductor (approximately 3,400 turns, 800 mH and 63 Ω)

• 6 V battery


• voltage probe and current probe

• two 75 Ω resistors (or close in value to resistance of inductor)


• knife-edge switch

RL

L



Activity 2-1: Inductors in switching circuits.

+

-

Vbatt

L

R

CPB

+

-

VPA

S

+

-

Figure 6

Consider the circuit in Figure 6. The ‘lozenge’ shape

represents the real coil you are using, which we model as an

ideal inductor in series with a resistor.

Question 2-1: Redraw the circuit (in the space to the right of

the figure), replacing the coil with an ideal inductor in series

with a resistor. Label all values. Be sure that VPA is shown

across the inductor/associated resistance combination (but not

across the “75 Ω” resistor).

1. Before hooking up the circuit, use the multimeter to

measure the resistance of your inductor, the resistor, the

inductance of the inductor, and the voltage of the battery.

Resistance of resistor: R = _____________ Ω

Resistance of inductor: L

R = _____________ Ω

Inductance of inductor: L = _____________ mH

Battery voltage: batt

V = _____________ V

Prediction 2-1: Calculate the expected current through CPB

and the voltage VPA after the switch has been closed for a long

time (show your work):

CPB current: ______________________

VPA voltage: ______________________



In Investigation 1 you observed that a changing magnetic field

inside an inductor results in an induced voltage across the

inductor. You also observed that a current through the coil

causes a magnetic field. Therefore a changing current through

an inductor will induce a voltage across the coil itself, and this

voltage will oppose (but not prevent!) the change.

Prediction 2-2: Calculate the expected current through CPB

and the voltage VPA at the instant just after when the switch is

closed (show your work):

CPB current: ______________________

VPA voltage: ______________________

vo

ltag

e,

VP

A

cu

rren

t, C

PB

open closedopenopen closedclosed

Prediction 2-3: On the axes above, sketch your qualitative

prediction for the current through CPB and the voltage

across VPA as switch S goes from open to closed to open

etc., several times.. [Hint: Does the voltage VPA decay all

the way to zero after the switch has been closed for a long

time? What if it were connected across an ideal (zero

resistance) inductor?]

2. Connect the circuit in Figure 6, and open the experiment

file called L07A2-1 Switched LR Circuit. Use a knife-



edge switch (momentary contact type switches tend to

“bounce”).

3. Measure the current and voltage as the switch is closed and

opened, keeping it closed or opened for about a second

each time.


You should observe the current rising to its maximum value as

follows:

( )max1 tI I e τ−

= −

where the time constant

L Rτ =

is the time it takes the current to reach about 63% (actually

1 - 1/e) of its final value.

Question 2-2: What value should you use for R?

5. Based on your redrawn circuit in step 2, calculate the

expected time constant.

L _____________

totalR _____________

predτ _______________ milliseconds

Now use the Smart Tool to measure the maximum current

on your graph, and the time it takes to reach 63% of that

maximum. You will have to spread out the time scale.

expτ _________________milliseconds.



6. Replace the inductor by a resistor of (at least

approximately) a value equal to the resistance of the

inductor. Take data again, opening and closing the switch.


Question 2-3: Is there a fundamental difference between

inductors and resistors? Explain.

Activity 2-2: Inductors in Switching Circuits, Modified

You may have noticed in the previous circuit that, when the

switch is opened the current decrease does not follow the

normal L/R time constant. By opening the switch we are

attempting to cut off the current instantaneously. This causes

the magnetic field to rapidly collapse. Such a rapid change in

the flux will induce a correspondingly large voltage. The

voltage will increase until either the air breaks down (you can

sometimes see or hear the tiny sparks). [In fact, if your tender

fingers are wee bit too close, you may find yourself making an

odd yelping sound.]

Figure 7

To remedy this, we will modify the circuit (Figure 7) so as to

give the current somewhere to go. Note that the circuit is

essentially the same as that for Activity 2-1, except that an

L

+

-V

S1

2

5

4

1

6

R

CPB+-

VPA

+

-

S2

3

Rinternal



extra wire and another switch (S2) have been added. We have

also explicitly shown the battery's internal resistance as we will

need to consider its effects.

We will now keep switch S1 closed during data taking. Its

purpose will be to prevent the battery from running down when

data are not being collected, so use the knife-edge switch

here. It is switch S2 that we will be opening and closing during

data taking.

For the following discussions we will assume switch S1 is

always closed (connected) when taking data. However, switch

S1 should be open (disconnected) when data are not being

collected.

The figure on the left below shows the equivalent circuit

configuration for Figure 7 when switch S2 is open (remember,

switch S1 is closed during data taking). In that case we

assumed that Rinternal << R1 and so we could safely ignore it.

Question 2-4: In the space on the right above, draw the

equivalent circuit configuration when switch S2 is closed (S1 is

also closed). NOTE: In this case, we cannot ignore Rinternal. In

fact, this time we will assume that Rinternal is much larger than

the resistance of the wires and the switches. [Don’t forget to

replace the real inductor with the ideal inductor/internal

resistance model.]

Don’t forget: We observe the voltage across the real inductor;

We cannot observe the voltages across the internal resistor or

ideal inductor alone.

Because the voltage induced across the inductor opposes an

instantaneous change in current, the current flow through the

inductor just after S2 is closed must be the same as the current

flow through it just before S2 is closed. [If not, there would

have been an instantaneous change in current, which cannot

happen.)]

L +

- V

21

R1

CPB+ -

VPA

+

-



Table 2-1

S2 has been

open for a

long time

Just after

S2 is closed S2 has been

closed for a

long time

Just after

S2 is open

Current in CPB: (clockwise, zero or

counterclockwise)

Induced voltage (V6>V5, V5>V6 or

V6=V5)

Prediction 2-4a: Suppose that S2 has been open for a long

time. In the first column of Table 2-1, predict the current in

the circuit just before S2 is closed. Now predict in the second

column of the table the current just after S2 is closed. Similarly,

predict the current in the circuit just before S2 is opened (when

S2 has been closed for a long time). Now predict in the fourth

column the current just after S2 is opened. Discuss your

reasoning.

Prediction 2-4b: Now consider the voltage across the

inductor, L. Based on your predictions for current, will the

potential at ‘6’ be greater than, less than or equal to the

potential at ‘5’ just after S2 is closed? Write your prediction in

the second row of Table 2-1 and explain below.



Prediction 2-4c: Will the potential at ‘6’ be greater than, less

than or equal to the potential at ‘5’ just after switch S2 is

opened? Write your prediction in Table 2-1 and explain

below.

Prediction 2-5: On the axes below, sketch your qualitative

predictions for the induced voltage across the inductor and

current through the circuit for each of the four time intervals.

[Hint: recall that the voltage across an inductor can change

almost instantaneously, but the current through the inductor

cannot change instantaneously. The induced voltage opposes

an instantaneous change in current and, thus, the change in

current must take place relatively slowly.]

vo

ltag

e, V

PB

Acu

rren

t, C

PB

S2 open S2 closed S2 open S2 closed

(1) (1)(4)(3)(2) (3)(2)

just after

S2 closed

just after

S2 closedjust after

S2 opened



Test your prediction.

1. Connect the circuit shown in Figure 7. Use a knife-edge

switch for S1 and a contact telegraph switch for S2.

2. Open the experiment file L07A2-1 Switched LR Circuit if

it's not already open.

3. Close switch S1 and leave it closed for the rest of this step.

Measure the current CPB and voltage VPA by switching S2

open and closed. Each time you switch, hold the switch

open or closed for about a second.

4. Print your graph.

5. After you have collected your data, open switch S1. (This

saves the battery from completely discharging while you

are not using it.)

Question 2-5: Discuss how well your observations agree with

your predictions. Address these questions: Is the battery

voltage driven all the way to zero by the connection S2 across

it? Does the inductor's internal resistance have an observable

effect?

L08-1


Name _______________________ Date ____________ Partners________________________________

Lab 8 - INTRODUCTION TO AC

CURRENTS AND VOLTAGES

OBJECTIVES

• To understand the meanings of amplitude, frequency,

phase, reactance, and impedance in AC circuits.

• To observe the behavior of resistors, capacitors, and

inductors in AC circuits.

OVERVIEW

Until now, you have investigated electric circuits in which a

battery provided an input voltage that was effectively constant

in time. This is called a DC or Direct Current signal. [A

steady voltage applied to a circuit eventually results in a steady

current. Steady voltages are usually called DC voltages.]

Signals that change over time exist all around you and many of

these signals change in a regular manner. For example, the

electrical signals produced by your beating heart change

continuously in time.

L08-2 Lab 8 - AC Currents & Voltage


There is a special class of time-varying signals. These signals

can be used to drive current in one direction in a circuit, then in

the other direction, then back in the original direction, and so

on. They are referred to as AC or Alternating Current signals.

time

volt

age

Examples of AC Signals

volt

age

time

It can be shown that any periodic signal can be represented as a

sum of weighted sines and cosines (known as a Fourier series).

It can also be shown that the response of a circuit containing

resistors, capacitors, and inductors (an “RLC” circuit) to such a

signal is simply the sum of the responses of the circuit to each

sine and cosine term with the same weights.

We further note that a cosine is just a sine that is shifted in time

by one-quarter cycle. So, to analyze an RLC circuit we need

only find the response of the circuit to an input sine wave of

arbitrary frequency.

Let us suppose that we have found a way to generate a current

of the form:

( )max( ) sinI t I tω= (1)

Note: Here we use the angular frequency, ω, which has units

of radians per second. Most instruments report the frequency, f, which has units of cycles per second or Hertz (Hz). The

frequency is the inverse of the period ( 1f T= ). Clearly,

2 fω π= .

volt

age

volt

age

time

time

Examples of Time-Varying Signals

L08-3


We can see from Ohm’s Law that the voltage across a resistor is then given:

( ) ( )max ,( ) sin sinR R max

V t I R t V tω ω= = (2)

Without proof1 we will state that the voltage across a capacitor

is given by:

( )( ) cos sin2

maxC C,max

IV t t V t

C

πω ω

ω

= − = −

(3)

and the voltage across an inductor is given by:

( ) ( ) ,cos sin2

L max L maxV t LI t V t

πω ω ω

= = +

(4)

Figure 1 shows a plot of the phase relationships between I, R

V ,

CV , and

LV . We can see that the voltage across a resistor is in

phase with the current; the voltage across an inductor leads the

current by 90° ; and the voltage across a capacitor lags the

current by 90° .

Figure 1

We make the following definitions,

1

CX

Cω

≡ and LX Lω≡ (5)

CX is called the reactance of the capacitor and

LX is the

reactance of the inductor. Capacitors and inductors behave like frequency dependent resistances, but with the additional

effect of causing a 90± ° phase shift between the current and

the voltage.

1 You can verify these equations by plugging them into C

V q C= (or CdV

I Cdt

= ) and L

dIV L

dt= .



Arbitrary combinations of resistors, capacitors and inductors will also have voltage responses of this form (a generalized

Ohm’s Law):

sin( )max

V I Z tω ϕ= + (6)

Z is called the impedance (and has units of resistance, Ohms)

and ϕ is called the phase shift (and has units of angle, degrees

or radians). The maximum voltage will be given by:

max maxV I Z= (7)

For a capacitor, 1C

Z X Cω= = and 90ϕ = − ° while for an

inductor, L

Z X Lω= = and 90ϕ = + ° .

Figure 2 shows the relationship between V and I for an

example phase shift of +20°. We say that V leads I in the

sense that the voltage rises through zero a time t∆ before the

current. When the voltage rises through zero after the current,

we say that it lags the current.

Figure 2

The relationship between ϕ and t∆ is given by

2 360t

f tT

ϕ π∆

= = °× ∆

(8)

where T is the period and f is the frequency.

In Investigation 1, you will explore how a time-varying signal

affects a circuit with just resistors. In Investigations 2 and 3, you will explore how capacitors and inductors influence the

current and voltage in various parts in an AC circuit.

L08-5


INVESTIGATION 1: AC SIGNALS WITH RESISTORS

In this investigation, you will consider the behavior of resistors in a circuit driven by AC signals of various frequencies.



• 500 Ω “reference” resistor

• 1 kΩ “device under test” resistor

• multimeter

• test leads

• external signal generator

Activity 1-1: Resistors and Time-Varying (AC) Signals.

VR

VPA

RVPB

V

+

_

R0

V0

VPC

+

+

_

_

Figure 3

Consider the circuit in Figure 3. This configuration is known

as a voltage divider and is a very commonly used circuit element. When implemented with resistors, it is used to

attenuate signals (mechanical volume controls for stereos are usually adjustable voltage dividers). As we’ll see later, when

implemented with capacitors and/or inductors, the resulting frequency dependent attenuator can be used to separate high

frequencies from low ones. Such circuits are called filters.

Resistor R0 is our “reference” resistor (and how we’ll measure

the current) and R is our “device under test” (or DUT, if you like TLA’s

2).

We seek to find the relationship between the driving voltage

V , and the voltage across our “device under test”, R

V . To do

so we use Kirchoff’s Circuit Rules and Ohm’s Law.

2 A TLA is a “Three Letter Acronym”, and they are all too common in technical jargon.



First, we see from the Junction Rule that the current must be the same in both resistors. Second, from the Loop Rule we see

that the sum of the emf’s and voltage drops must be zero. Hence:

0 RV V V= +

Then using Ohm’s Law we get:

( )0 0V IR IR I R R= + = +

Plugging in our standard forms for the current and voltage [Equations (1) and (6)], we get

max max 0sin( ) ( )sin( )V t I R R tω ϕ ω+ = +

Recall the trigonometric identity:

sin( ) sin( )cos( ) cos( )sin( )α β α β α β+ = + (9)

Hence:

( ) ( ) ( ) ( ) ( ) ( )0sin cos cos sin sin

max maxV t t I R R tω φ ω φ ω+ = +

Equating the coefficients of ( )sin tω and ( )cos tω yields

( ) ( )0cosmax maxV I R Rφ = +

and

( )sin 0maxV φ =

Since ( )sin 0ϕ = and ( )cos ϕ is positive, there is no phase

shift:

0ϕ = (10)

The magnitude of the current is given by:

( )0max maxI V R R= + (11)

Hence, the impedance of this series combination of two

resistors is given by3:

0RR max max

Z V I R R= = + (12)

We can now calculate the magnitude of the voltage across R:

,R max max max

RR

RV I R V

Z= = (13)

Similarly,

0 00, 0

0

max max max max

RR

R RV I R V V

Z R R

= = =

+ (14)

3 The subscript “RR” simply serves to remind us that this Z for two resistors in series.

L08-7


1. Measure the resistors:

R0: __________ R: __________

2. Connect the circuit in Figure 3.

NOTE: Use the red connector (“Lo ?”) on the power supply

for the positive (“+”) side and the black connector for the

negative terminal.

Question 1-1: Assume that V is a sinusoid of amplitude (peak

voltage) 4Vmax

V = . Use Equations (13) and (10) to predict

RRZ ,

,R maxV and ϕ for driving frequencies of 50 Hz, 200 Hz,

and 800 Hz. Show your work.

f = 50 Hz RR

Z = ,R max

V = ϕ =

f = 200 Hz RR

Z = ,R max

V = ϕ =

f = 800 Hz RR

Z = ,R max

V = ϕ =

Question 1-2: On the axes below, sketch your quantitative

predictions for V, V0 and VR, versus time, t. Assume that V is a

200 Hz sinusoid of amplitude (peak voltage) 4 V. Draw at least one full cycle. Show your work below.



3. Turn on the signal generator (the switch is on the back). Set the frequency to 200 Hz, the amplitude to about half

scale, and select a sinusoidal waveform (the other options are “triangle” and “square” waves).

4. Open the experiment file called L08A1-1 AC Voltage

Divider. You will see an oscilloscope display. An

oscilloscope is a device that shows voltages versus time. Each voltage waveform is called a trace and you should see

three traces on the screen (VPA, VPB, and VPC).

5. Take a little to play with the controls. Click Start. Then

click on the little black arrows for the time and voltage scales. See how they change the display.

6. Play with the “trigger level” (the little arrow on the left-hand-side) a bit to see how it operates. The trigger

determines when oscilloscope starts its sweep by looking at a specific input and determining when it either rises above

or falls below a specified level. You can select the “trigger source” by clicking on the appropriate box on the right-

hand-side of the scope display. You can “click-and-drag” the trigger level with the mouse. Note: The trigger level

indicator will be the same color as trigger source trace.

7. Double-click on the display to bring up the settings

window. Here you can select the trigger source, set the trigger level, and the direction that the signal must be

passing the trigger level to start the scope.

8. Verify that the signal generator is set to 200 Hz and adjust

the amplitude until Vmax (VPC) is 4 V. Set the oscilloscope such that the time axis is set to one millisecond per division

and that all three voltage scales are set to one volt per division. [Note that the frequency knob is “speed

L08-9


sensitive” in that the faster you spin the dial, the more it changes the frequency per unit angle.]

9. When you have a good display, click stop.

10. Use the Smart Tool to find the maxima for V, V0, and VR.

Also measure the time delay (∆t) between V and V0. Enter these data into the middle column of Table 8-1. Note that

you’ll get more precise time measurements if you look at where the traces “cross zero”.

Caution: When using the Smart Tool, make sure that it is “looking” at the correct trace. The digits will be the same color

as the trace.

11. Vary the frequency between about 20 Hz and 1,000 Hz.

Question 1-3: Describe your observations, with particular

attention paid to the amplitudes and relative phases.

12. Now make the same measurements for 50 Hz and 800 Hz and enter the data into Table 8-1.

13. Calculate max 0,max 0

I V R= and RR max max

Z V I= for each of

frequency. Also calculate the phase shifts (in degrees) between V and V0, V and VR, and, between V0 and VR.

Enter the results into Table 8-1.



Table 8-1

Question 1-4: Discuss the agreement between your

experimental results (see Question 1-3 and Table 8-1) and your predictions. Specifically consider the frequency dependences

(if any).

Note: Do not disconnect this circuit as you will be using a

very similar one in Investigations 2 and 3.

INVESTIGATION 2: AC SIGNALS WITH CAPACITORS

In this investigation, you will consider the behavior of capacitors in a circuit driven by AC signals of various

frequencies.




• 470 nF “device under test” capacitor

• multimeter

• test leads

• signal generator

f = 50 Hz f = 200 Hz f = 800 Hz

Vmax

V0,max

VR,max

∆t (V-V0)

Imax

ZRR

φ (V-V0)

L08-11


Activity 2-1: Capacitors and Time-Varying (AC) Signals.

VC

VPA

C VPB

V

+

_

R0

V0

VPC

Figure 4

Consider the circuit in Figure 4. Resistor R0 is again our

“reference” resistor. C is our “device under test”.

Now we seek to find the relationship between the driving

voltage, V , and the voltage across the capacitor, C

V . From

Kirchhoff’s Rules and Equations (2) and (3) we get:

( ) ( ) ( )0sin sin cosmax max max C

V t I R t I X tω ϕ ω ω+ = −

Once again using the trigonometric identity and equating the

coefficients of ( )sin tω and ( )cos tω , we get

( ) 0cosmax max

V I Rφ =

and

( )sinmax max C

V I Xφ = −

Hence the phase shift is given by

( )tan CX

Rϕ = − (15)


( )2 2 2 2

0max max CI V R X= + (16)

Hence, the impedance of this series combination of a resistor

and a capacitor is given by:

2 2

0RC max max CZ V I R X= = + (17)

We can now calculate the magnitude of the voltage across C:

,C

C max max C max

RC

XV I X V

Z= = (18)



Similarly,

00, 0max max max

RC

RV I R V

Z= = (19)

14. Measure C:

C: __________



voltage) 4Vmax


Z ,,C max

V and ϕ for driving frequencies of 50 Hz, 200 Hz, and

800 Hz. Show your work. [Hint: Don’t forget that

1 1 2C

X C fCω π= = !]

f = 50 Hz RCZ = ,C max

V = ϕ =


V = ϕ =


V = ϕ =


predictions for V, V0 and VC versus time, t. Assume that V is a

200 Hz sinusoid of amplitude (peak voltage) 4 V. Draw at

least one full cycle. Show your work below.

L08-13


16. Continue to use L08A1-1 AC Voltage Divider.


the amplitude until Vmax (VPC) is 4 V. Set the oscilloscope

such that the time axis is set to one millisecond per division

and that all three voltage scales are set to one volt per

division.

18. Click Start. Trigger on V0 (as it is proportional to the

current). When you have a good display, click stop.

19. Use the Smart Tool to find the maxima for V, V0, and VC.

Also measure the time delay (∆t) between V and V0. Enter

these data into the middle column of Table 8-2.




21. Now make the same measurements for 50 Hz and 800 Hz

and enter the data into Table 8-2.




I V R= and RC max max

Z V I= for each

frequency. Also calculate the phase shifts (in degrees)

between V and V0. Enter the results into Table 8-2.

Table 8-2


experimental results (see Question 2-3 and Table 8-2) and

your predictions. Specifically consider the frequency

dependences (if any).

INVESTIGATION 3: AC SIGNALS WITH INDUCTORS

In this investigation, you will consider the behavior of

inductors in a circuit driven by AC signals of various

frequencies.




• 800 mH “device under test” inductor

• multimeter

f = 50 Hz f = 200 Hz f = 800 Hz

Vmax

V0,max

VC,max

∆t (V-V0)

Imax

ZRC

φ (V-V0)

L08-15


• test leads

• signal generator

Activity 3-1: Inductors and Time-Varying (AC) Signals.

VL

VPA

L VPB

V

+

_

R0

V0

VPC

Figure 5

Consider the circuit in Figure 5. Resistor R0 is again our

“reference” resistor. L is our “device under test”.

Now we seek to find the relationship between the driving

voltage, V , and the voltage across the inductor, L

V .

From Kirchoff’s Rules and Equations (2) and (4) we see:

( ) ( ) ( )0sin sin cosmax max max LV t I R t I X tω ϕ ω ω+ = +

[Remember that L

X Lω= .]

Using the identity and equating the coefficients of ( )sin tω and

( )cos tω , we get

( ) 0cosmax maxV I Rφ =

and

( )sinmax max LV I Xφ =

Hence the phase shift is given by:

( )

0

tan LX

Rϕ = (20)


( )2 2 2 2

0max max LI V R X= + (21)



Hence, the impedance of this series combination of a resistor

and an inductor is given by:

2 2

0RL max max LZ V I R X= = + (22)

We can now calculate the magnitude of the voltage across L:

, 2 2

0

L LL max max L max max

RL L

X XV I X V V

Z R X

= = =

+

(23)

Similarly,

0 0

0, 0 2 2

0

max max max max

RL L

R RV I R V V

Z R X

= = =

+

(24)

23. Measure L:

L: __________



voltage) 4Vmax


RLZ ,

,L maxV and ϕ for driving frequencies of 50 Hz, 200 Hz,

and 800 Hz. Show your work.

f = 50 Hz RL

Z = ,L max

V = ϕ =

f = 200 Hz RL

Z = ,L max

V = ϕ =

f = 800 Hz RL

Z = ,L max

V = ϕ =

L08-17



predictions for V, V0 and VL versus time, t. Assume that V is a

200 Hz sinusoid of amplitude (peak voltage) 4 V. Draw two

periods and don’t forget to label your axes. Show your work

below.

25. Continue to use L08A1-1 AC Voltage Divider.


the amplitude until Vmax (VPC) is 4 V. Set the oscilloscope

such that the time axis is set to one millisecond per division

and that all three voltage scales are set to one volt per

division.

27. Click Start. When you have a good display, click stop.

28. Use the Smart Tool to find the maxima for V, V0, and VL.

Also measure the time delays (∆t’s) between V and V0, V

and VL, and, between V0 and VL. Enter these data into the

middle column of Table 8-3.






30. Now make the same measurements for 50 Hz and 800 Hz

and enter the data into Table 8-3.


I V R= and RL max max

Z V I= for each

frequency. Also calculate the phase shifts (in degrees).

Enter the results into Table 8-3.

Table 8-3

f = 50 Hz f = 200 Hz f = 800 Hz

Vmax

V0,max

VL,max

∆t (V-V0)

Imax

ZRL

φ (V-V0)

L08-19



experimental results (see Question 3-3 and Table 8-3) and your

predictions. Specifically consider the frequency dependences

(if any).

WRAP-UP

Question 1: Do your results make intuitive sense for low

frequencies? Explain. Answer this by considering switched

DC circuits in the steady state (i.e., after things settle down).

[A DC current (or voltage) can be thought of as the limit of a

cosine as the frequency goes to zero.]

Question 2: Do your results make intuitive sense for high

frequencies? Explain. Answer this by considering switched

DC circuits immediately after the switch is closed or opened.



L09-1


Name ________________________ Date ____________ Partners_______________________________

Lab 9 –AC FILTERS AND RESONANCE

OBJECTIVES

• To understand the design of capacitive and inductive filters

• To understand resonance in circuits driven by AC signals

OVERVIEW

In a previous lab, you explored the relationship between

impedance (the AC equivalent of resistance) and frequency for

a resistor, capacitor, and inductor. These relationships are very

important to people designing electronic equipment. You can

predict many of the basic characteristics of simple AC circuits

based on what you have learned in previous labs.

Recall that we said that it can be shown that any periodic signal

can be represented as a sum of weighted sines and cosines

(known as a Fourier series). It can also be shown that the

response of a circuit containing resistors, capacitors, and

inductors (an “RLC” circuit) to such a signal is simply the sum

of the responses of the circuit to each sine and cosine term with

the same weights.

Recall further that if there is a current of the form

( )max( ) sinI t I tω= (1)

flowing through a circuit containing resistors, capacitors and/or inductors, then the voltage across the circuit will be of the form

( ) ( )max sinV t I Z tω ϕ= + . (2)

Z is called the impedance (and has units of resistance, Ohms)

and φ is called the phase shift (and has units of angle, radians).

The peak voltage will be given by

max maxV I Z= . (3)

L09-2 Lab 9 - AC Filters & Resonance


Figure 1 shows the relationship between V and I for an

example phase shift of +20°. We say that V leads I in the

sense that the voltage rises through zero a time t∆ before the

current. When the voltage rises through zero after the current,

we say that it lags the current.

Figure 1

The relationship between ϕ and t∆ is given by

2 or 360t

f tT

ϕ π ϕ∆

= = °× ∆

(4)

where T is the period and f is the frequency.

For a resistor, RZ R= and there is no phase shift ( 0

Rϕ = ). For

a capacitor, 1C C

Z X Cω= = and 90C

ϕ = − ° while for an

inductor, L LZ X Lω= = and 90

Lϕ = + ° . In other words:

sin( )R max

V I R tω= (5)

cos( )C max C

V I X tω= − (6)

and

cos( )L max L

V I X tω= (7)

XC is called the capacitive reactance and XL is called the

inductive reactance.

Let us now consider a series combination of a resistor, a

capacitor and an inductor shown in Figure 2. To find the

impedance and phase shift for this combination we follow the

procedure we established before.

V

L

C

R

Figure 2

Lab 9 - AC Filters & Resonance L09-3


From Kirchhoff’s loop rule we get:

R L CV V V V= + + (8)

Adding in Kirchhoff’s junction rule and Equations (2), (5), (6),

and (7) yields

( ) ( ) ( ) ( )sin sin cosmax RLC max L C

V t I R t X X tω ϕ ω ω+ = + −

Once again using a trigonometric identity1 and equating the

coefficients of ( )sin tω and ( )cos tω , we get

( )cosmax RLC max

V I Rφ =

and

( ) ( )sinmax RLC max L C

V I X Xφ = −

Hence the phase shift is given by

( )tan L C

RLC

X X

Rϕ

−

= (9)

and the impedance of this series combination of a resistor, an

inductor, and a capacitor is given by:

( )22

RLC max max L CZ V I R X X= = + − (10)

The magnitudes of the voltages across the components are then

,R max max max

RLC

RV I R V

Z= = (11)

,

LL max max L max

RLC

XV I X V

Z= = (12)

and

,

C

C max max C max

RLC

XV I X V

Z= = (13)

Explicitly considering the frequency dependence, we see that

( )

,22 1

R max max

RV V

R L Cω ω

=

+ −

(14)

( )

,22 1

L max max

LV V

R L C

ω

ω ω

=

+ −

(15)

and

( )

,22

1

1C max max

CV V

R L C

ω

ω ω

=

+ −

(16)

1 sin( ) sin( )cos( ) cos( )sin( )α β α β α β+ = +



This system has a lot in common with the forced mechanical

oscillator that we studied in the first semester. Recall that the

equation of motion was

F ma bv kx mx bx kx= + + = + +ɺɺ ɺ (17)

Similarly, Equation (8) can be written as

1

V Lq Rq qC

= + +ɺɺ ɺ (18)

We see that charge separation plays the role of displacement,

current the role of velocity, inductance the role of mass

(inertia), capacitance (its inverse, actually) the role of the

spring constant, and resistance the role of friction. The driving

voltage plays the role of the external force.

As we saw in the mechanical case, this electrical system

displays the property of resonance. It is clear that when the

capacitive and inductive reactances are equal, the impedance is

at its minimum value, R . Hence, the current is at a maximum

and there is no phase shift between the current and the driving

voltage.

Denoting the resonant frequency as LCω and the common

reactance of the capacitor and inductor at resonance as LCX ,

we see that, at resonance

( ) ( )LC C LC L LCX X Xω ω≡ =

so

1LC

LCω = (19)

and

LC

X L C= (20)

At resonance the magnitude of the voltage across the capacitor

is the same as that across the inductor (they are still 180° out of

phase with each other and ±90° out of phase with the voltage

across the resistor) and is given by

( ) ( ), ,LC

C max LC L max LC max

XV V V

Rω ω= = (21)



In analogy with the mechanical case, we call the ratio of the

amplitude of the voltage across the capacitor (which is

proportional to q , our “displacement”) at resonance to the

driving amplitude the resonant amplification, which we denote

as Q,

( ),C max LC maxQ V Vω≡ (22)

Hence,

LC

Q X R L C R= = (23)

Figure 3 (below) shows the voltage across a capacitor

(normalized to the driving voltage) as a function of frequency

for various values of Q .

Figure 3

In this lab you will continue your investigation of the behavior

of resistors, capacitors and inductors in the presence of AC

signals. In Investigation 1you will explore the relationship

between peak current and peak voltage for a series circuit

composed of a resistor, inductor, and capacitor. You will also

explore the phase difference between the current and the

voltage. This circuit is an example of a “resonant circuit”. The

phenomenon of resonance is a central concept underlying the

tuning of a radio or television to a particular frequency.

INVESTIGATION 1: THE SERIES RLC RESONANT (TUNER) CIRCUIT

In this investigation, you will use your knowledge of the

behavior of resistors, capacitors and inductors in circuits driven

by various AC signal frequencies to predict and then observe

the behavior of a circuit with a resistor, capacitor, and inductor

connected in series.



The RLC series circuit you will study in this investigation

exhibits a “resonance” behavior that is useful for many familiar

applications such a tuner in a radio receiver.


• Voltage probes • Multimeter

• 510 Ω resistor • test leads

• 800 mH inductor • 820 nF capacitor

Consider the series RLC circuit shown in Figure 4 (below).

[For clarity, we don’t explicitly show the voltage probes.]

V

L

C

R

Figure 4

Prediction 2-1: At very low signal frequencies (less than

10 Hz), will ,R maxI and

,R maxV

be relatively large, intermediate

or small? Explain your reasoning.

Prediction 2-2: At very high signal frequencies (well above

3,000 Hz), will the values of ,R maxI and

,R maxV be relatively

large, intermediate or small? Explain your reasoning.



1. On the axes below, draw qualitative graphs of CX vs.

frequency and LX vs. frequency. Clearly label each curve.

Frequ ency

X L

X C

and

2. On the axes above (after step 1) draw a curve that

qualitatively represents L CX X− vs. frequency. Be sure to

label it.

3. Recall that the frequency at which Z is a minimum is

called the resonant frequency, LCf and that the common

reactance of the inductor and the capacitor is LCX . On the

axes above, mark and label LCf and LC

X .

Question 2-1 At LCf will the value of the peak current, max

I ,

in the circuit be a maximum or minimum? What about the

value of the peak voltage, ,R maxV , across the resistor? Explain.

4. Measure the 510 Ω resistor (you have already measured the

inductor and the capacitor):

R : __________

5. Use your measured values to calculate the resonant

frequency, the reactance of the capacitor (and the inductor)



at resonance, and the resonant amplification factor. Show

your work. [Don’t forget the units!]

LCf : __________

LCX : __________

Q : __________

Activity 2-1: The Resonant Frequency of a Series RLC

Circuit.

1. Open the experiment file L09A2-1 RLC Filter.

2. Connect the circuit with resistor, capacitor, inductor and

signal generator shown in Figure 4. [Use the internal

generator.]

3. Adjust the generator to make a 50 Hz signal with amplitude

of 2 V.

4. Connect voltage probe VPA across the resistor, VPB across

the inductor, and VPC across the capacitor.

5. Use the Smart Tool to determine the peak voltages

(,R maxV ,

,L maxV , and ,C maxV ). Enter the data in the first row of

Table 2-1.

6. Repeat for the other frequencies in Table 2-1.

Table 2-1

f (Hz) ,R maxV (V) ,L maxV (V)

,C maxV (V)

50

100

200

400

800



7. Measure the resonant frequency of the circuit to within a

few Hz. To do this, slowly adjust the frequency of the

signal generator until the peak voltage across the resistor is

maximal. [Use the results from Table 2-1 to help you

locate the resonant frequency.]

,LC expf : __________

Question 2-2: Discuss the agreement between this

experimental value for the resonant frequency and your

calculated one.

8. Use the Smart Tool to determine the peak voltages at

resonance.

maxV : __________

,R maxV : __________

,L maxV : __________

,C maxV : __________

Question 2-3: From these voltages, calculate Q and discuss

the agreement between this experimental value and your

calculated one.



Question 2-4: Calculate your experimental value of LCX and

discuss the agreement between this value and your calculated

one.

Prediction 2-5: What will we get for Q if we short out the

resistor? Show your work.

9. Short out the resistor.

10. Measure Q . [You may have to lower the signal voltage to

0.5 V.] Show your work. Explicitly indicate what you had

to measure.

Q : __________

Question 2-6: Discuss the agreement between this

experimental value and your predicted one.



Activity 2-2: Phase in an RLC Circuit

In previous labs, you investigated the phase relationship

between the current and voltage in an AC circuit composed of

a signal generator connected to one of the following circuit

elements: a resistor, capacitor, or an inductor. You found that

the current and voltage are in phase when the element

connected to the signal generator is a resistor, the current leads

the voltage with a capacitor, and the current lags the voltage

with an inductor.

You also discovered that the reactances of capacitors and

inductors change in predictable ways as the frequency of the

signal changes, while the resistance of a resistor is constant –

independent of the signal frequency. When considering

relatively high or low signal frequencies in a simple RLC

circuit, the circuit element (either capacitor or inductor) with

the highest reactance is said to “dominate” because this

element determines whether the current lags or leads the

voltage. At resonance, the reactances of capacitor and inductor

cancel, and do not contribute to the impedance of the circuit.

The resistor then is said to dominate the circuit.

In this activity, you will explore the phase relationship between

the applied voltage (signal generator voltage) and current in an

RLC circuit.

Consider again our RLC circuit (it is the same as Figure 4).

V

L

C

R

Figure 5

Question 2-7: Which circuit element (the resistor, inductor, or

capacitor) dominates the circuit at frequencies well below the

resonant frequency? Explain.



Question 2-8: Which circuit element (the resistor, inductor, or

capacitor) dominates the circuit at frequencies well above the

resonant frequency? Explain.

Question 2-9a: In the circuit in Figure 5, will the current

through the resistor always be in phase with the voltage across

the resistor, regardless of the frequency? Explain your

reasoning.

Question 2-9b: If your answer to Question 2-9a was no, then

which will lead for frequencies below the resonant frequency

(current or voltage)? Which will lead for frequencies above the

resonant frequency (current or voltage)?

Question 2-10a: In the circuit in Figure 5, will the current

through the resistor always be in phase with applied voltage

from the signal generator? Explain your reasoning.

Question 2-10b: If your answer to Question 2-10a was no,

then which will lead for frequencies below the resonant



frequency (current or voltage)? Which will lead for

frequencies above the resonant frequency (current or voltage)?

1. Continue to use L09A2-1 RLC Filter.

2. Reconnect the circuit shown in Figure 5. Connect voltage

probe VPA across the resistor, VPB across the inductor, and

VPC across the capacitor.

3. Start the scope and set the signal generator to a frequency

20 Hz below the resonant frequency you measured in

Investigation 2, and set the amplitude of the signal to 2 V.

Question 2-11: Which leads – applied voltage, current or

neither – when the AC signal frequency is lower than the

resonant frequency? Discuss agreement with your prediction.

4. Set the signal generator to a frequency 20 Hz above the

resonant frequency.

Question 2-12: Which leads – applied voltage, current or

neither – when the AC signal frequency is higher than the

resonant frequency? Discuss agreement with your prediction.



Question 2-13: At resonance, what is the phase relationship

between the current and the applied voltage?

5. Use this result to find the resonant frequency.

,LC phasef : __________

Question 2-14: Discuss how this experimental value compares

with your calculated one.

Question 2-15: How does this experimental value for the

resonant frequency compare with the one you determined by

looking at the amplitude? Comment on the relative

“sensitivities” of the two techniques.

L10-1


Name Date Partners

Lab 10 - GEOMETRICAL OPTICS

OBJECTIVES

• To examine Snell’s Law.

• To observe total internal reflection.

• To understand and use the lens equations.

• To find the focal length of a converging lens.

• To discover how lenses form images.

• To observe the relationship between an object and the image formed by a lens.

• To discover how a telescope works.

OVERVIEW

Light is an electromagnetic wave. The theory of the propagation

of light and its interactions with matter is by no means trivial;

nevertheless, it is possible to understand most of the fundamental

features of optical instruments such as eyeglasses, cameras,

microscopes, telescopes, etc. through a simple theory based on the

idealized concept of a light ray.

A light ray is a thin “pencil” of light that travels along a straight

line until it encounters matter, at which point it is reflected,

refracted, or absorbed. The thin red beam from a laser pointer is a

good approximation of such a ray. The study of light rays leads to

two important experimental observations:

1. When a light ray is reflected by a plane surface, the angle of

reflection θ2 equals the angle of incidence θ1, as shown in

Figure 1.

θ2

θ1

Figure 1 Reflection: θ1 = θ2

2. When a light ray travels from one transparent medium into

another, as shown in Figure 2, the ray is generally “bent”

L10-2 Lab 10 - Geometrical Optics


(refracted). The directions of propagation of the incident and

refracted rays are related to each other by Snell’s Law:

1 1 2 2sin sinn nθ θ= (1)

where the dimensionless number n is called the index of refraction and is characteristic of the material.

θ1

n1 n2

θ2

Figure 2 Refraction: 1 1 2 2sin sinn nθ θ=

Note that if 1 1 2sinn nθ > , no solution is possible for 2sinθ . In this

case, none of the light will pass through the interface. All of the light will be reflected. This total internal reflection is more

perfect than reflection by any metallic mirror.

Most transparent materials have indices of refraction between 1.3

and 2.0. The index of refraction of a vacuum is by definition unity. For most purposes, the index of refraction of air (nair = 1.0003) can

also be taken as unity.

Accurate measurements show the index of refraction to be a

function of the wavelength and thus of the color of light. For most materials, one finds that:

redblue

nn > (2)

A simplified theoretical explanation of these observations is given by Huygens’ Principle, which is discussed in elementary physics

texts.

Note: The room lights will be turned out for these investigations. It will sometimes be difficult to read and write in this manual. Use

the desk lamp as needed. Be patient!

Lab 10 - Geometrical Optics L10-3


INVESTIGATION 1: SNELL’S LAW

In this investigation, you will observe and verify Snell’s Law by using both a rectangular block and a prism. You will also observe

total internal reflection in a prism.


• Rectangular block made of Lucite

• Triangular prism made of Lucite

• Triangle

• Protractor

• Light ray box

• Graphing paper from roll (approximately 40 cm each for Activities 1-1 and 1-2).

Activity 1-1: Verifying Snell’s Law

In this activity, you will verify Snell’s Law by using the light ray box with a single ray and the rectangular plastic block.

PLEASE TAKE CARE NOT TO SCRATCH THIS BLOCK

OR THE OTHER OPTICAL ELEMENTS!

θ1

θ1'

θ2

θ2'

t

s

Figure 3 Plate with parallel surfaces.

From Figure 3 we can see that Snell’s Law and the symmetry of the geometry imply (assuming nair ≈ 1):

1 2

sin sinn θ θ= (3)

1 1'θ θ= (4)



and

( )1 2 2sin coss t θ θ θ= − (5)

1. Using the single aperture mask, let a single ray from the ray box fall on a piece of graph paper such that it is aligned with the

grid. [It may help to tape the paper to the table to keep it from moving.]

Note: Only one diagram will be drawn for each group. There are at least three activities that have you draw light rays, so make sure

each partner does at least one ray tracing diagram.

2. Place the block on the graph paper. Make sure that there is at least 10 cm of paper on either side of the block. Align the block

so that the light ray is incident at an oblique angle with the block (as in Figure 3). Trace the outline of the block on the

graph paper.

Hint: Larger values of θ1 produce better results.

3. Mark on the graph paper the entry and exit points of the light. Also mark points on the incident and exit rays far from block.

This will be necessary to determine the angles.

4. After removing the block, trace the light ray paths and label the

diagram.

5. Use the protractor to measure the angles θ1, θ2, θ1' and θ2

'.

Extrapolate the incident ray so that you can measure s, the “shift” (or offset) of the output ray relative to the incident ray.

Record your results in Table 1-1.

Note: We have used the subscript 1 for air and the subscript 2 for plastic, regardless of the direction of the ray. Other conventions

are equally valid.

Table 1-1

1θ 2θ

1θ ′

2θ ′ s

6. Determine the index of refraction n for the block. Show your work.

n



Question 1-1: Is Equation (4) satisfied? [In other words, are the incident and exit rays parallel?] Discuss.

Question 1-2: Is Equation (5) satisfied? What does it tell you

about the path of a ray through the center of a thin lens? [Hint: Imagine your Lucite block getting thinner and think about how the

offset between the incident and exit rays would change.]

Activity 1-2: Light Passing Through a Prism

In this activity, you will study the propagation of light through a prism, as well as observe total internal reflection.

Figure 4 Refraction and total reflection in a prism.

Two examples of light propagation in a prism are shown in

Figure 4. As you will recall, at each surface some of the light is reflected and some of the light is refracted. Figure 4a shows a light

ray entering the prism at A at an angle θin (relative to the normal), the refracted ray at the front surface, and the refracted ray at the

rear surface leaving the prism at B at an angle θout. [Note: At each

θin

AB

1refracted 1in

(a) (b)

θout

2reflected 2refracted

AθC

2in α C

D



interface there will also be a reflected ray, but, for clarity, we don’t show them here.]

Figure 4b shows the case where the refracted output ray would come out along the edge of the surface (θout = 90°). Any angle

smaller than θC will produce light that hits the rear surface so that

sin 1n α ≥ , the condition for total internal reflection.

1. To observe this total internal reflection, a triangular prism will be used. Place the prism on a clean area of the graph paper or a

new sheet.

2. Set the ray box so a single light ray falls on one side of the

prism.

3. Vary the entrance angle of the ray by slowly rotating the prism.

Note that there is a point at which no light is refracted out. Mark the positions of the rays when this total internal reflection

occurs, as well as trace around the prism. Make sure to mark the incident ray, the point at which this ray strikes the back of

the prism, and the reflected ray once it has exited the prism.

4. From these markings and using the protractor, find the internal

reflection angle α. [Hint: Extend lines AC and CD and measure the included angle, which is α + 90°]

α _______________________

Question 1-3: Use Snell’s Law to derive an equation for n in terms

of α. Show your work and calculate n.

n _______________________

5. Slowly rotate the prism again and note that the exiting ray spreads out into various colors just before total internal

reflection occurs.



Question 1-4: Why does the light spread into different colors prior to total internal reflection?

6. Start at the point of total internal reflection and rotate the prism

slightly to increase the entrance angle. You should see a weak ray just grazing along the outside of the prism base. Note very

carefully which color emerges outside the prism first (red or blue).

Question 1-5: Discuss what this tells you about the relative magnitudes of nred and nblue for Lucite.

INVESTIGATION 2: CONVERGING LENSES

Most optical instruments contain lenses, which are pieces of glass

or transparent plastics. To see how optical instruments function, one traces light rays through them. We begin with a simple

example by tracing a light ray through a single lens.

We apply Snell’s Law to a situation in which a ray of light, coming

from a medium with the refractive index n1 = 1, e.g. air, falls onto a glass sphere with the index n, shown in Figure 5.



F0

h

θ1 n

α

R

f0

x

θ2 y

Figure 5 Spherical lens.

We have in that case

1 2sin sinnθ θ= (6)

From Figure 5, we see

2sin siny x Rα θ= = (7)

or

2sin

sinx R

θ

α

= . (8)

If we make the simplifying assumption h << R, we can use the

approximation

( )2 2

sinh h

R xR x h

α = ≈

++ +

. (9)

This yields

12

( ) sin( )sin

R R xR xx R

h nh

θθ

++≈ =

Or, as 1sin h Rθ = ,

xRnx +≈ , (10)

where R + x is the distance from the front of the sphere to the point

F where the ray crosses the optical axis. We call this distance the focal length f0. Setting f0 = R + x, Equation (10) now reads:

( )0 0n f R f− ≈ , or 1

0−

≈

n

nRf (11)



This is a remarkable result because it indicates that, within the

limits of our approximation ( h << R ), the focal length 0f is

independent of h. This means that all rays that come in parallel to

each other and are close to the axis are collected in one point, the focal point, F0.

Note that our simple theory of a lens applies only to those cases in which the focal point is inside the sphere. A lens whose focal

point is on its inside is not very useful for practical applications; we want it to be on the outside. [Actually, whether inside or out,

spheres, for various reasons, do not make very useful lenses.] We will therefore study a more practical lens, the planoconvex lens.

A planoconvex lens is bounded on one side by a spherical surface with a radius of curvature R and on the other by a plane (see

Figure 6). To keep things simple we make the additional assumption that it is very thin, i.e. that d << R. Now we trace an

arbitrary ray that, after having been refracted by the spherical front surface, makes an angle θ1 with the optical axis, as shown in

Figure 6.

F0

h

θ1

n

α

R

f0

d

f

θ3

F

Figure 6 Focal point of planoconvex lens.

If there were still a full glass sphere, this ray would intersect the

optical axis at the point F0, a distance f0 from the front surface. On encountering the planar rear surface of the lens it will instead,

according to Snell’s Law, be bent to intercept the axis at the point F, a distance f from the front. Behind the rear surface is air,

so, on the encounter with the second surface Snell’s Law becomes:

3sin sinn α θ= (12)



But

3sinh

fθ ≈ and

0

sinh

fα ≈ (13)

Hence,

0ffn

≈ , (14)

i.e. in this case the distance f is independent of the distance h (as

long as h << R and d << R). Using Equation (11) in Equation (14)

we find that all incoming rays that are parallel to the optical axis of

a thin planoconvex lens are collected in a focal point at a distance

1−

=

n

Rf (15)

behind the lens.

What about rays that are not parallel? One can show that all rays

issuing from the same object point will be gathered in the same

image point (as long as the object is more than one focal length

away from the lens).

To find the image point, we only need consider two rays (we’ll

discuss three that are easy to construct) and find their intersection.

Let us assume that there is a point source of light at the tip of

object O at a distance o > f in front of the lens. Consider the three

rays issuing from this source shown in Figure 7:

f

i o

f

I

O

F F

1

3

2

Figure 7 Image construction.

1. A ray that is parallel to the axis. According to what we have

just learned, it will go through the focal point F behind the

lens.

2. A ray that goes through the focal point F in front of the lens.

With a construction analogous to the one shown in Figure 7, one

can show that light parallel to the axis coming from behind the



lens will go through the focal point in front. Our construction is

purely geometrical and cannot depend on the direction of the

light beam. We conclude that light that passes through the focal

point in front of the lens must leave the lens parallel to the axis.

This ray will intersect the first ray at the tip of image I at a

distance i behind the lens.

3. A ray that goes through the center of the lens. At the center,

the two glass surfaces are parallel. As we have seen, light

passing through such a plate will be shifted by being bent

towards the normal at the first interface and then back to the

original direction at the second interface. If the plate (in our

case the lens) is thin, the shift will be small. We assumed our

lens was very thin, so we can neglect any such shift.

From Figure 7 it should not be difficult for you to see (from

“similar triangles”) that:

I O I

o f

+= and

I O O

i f

+=

Hence we arrive at the following thin lens formulae:

iof

111+= (16)

O

o

I

i= (17)

We define magnification M to be the ratio of the image size I to

the object size O:

I

MO

≡ (18)

or [by application of Equations (16) and (17)]:

f

Mo f

=

−

(19)

The image in Figure 7 is called a real image because actual rays

converge at the image. The method of image construction used in

Figure 7, as well as thin lens formulae, can also be formally

applied to situations where that is not the case.

What about when the object is closer than one focal length? In

Figure 8, an object O is placed within the focal distance ( o f< ) of

a lens. Following the usual procedure, we draw the ray going

through the center of the lens and the one that is parallel to the

axis. We add a third ray, originating from O but going in a



direction as if it had come from the first focal point F. All these

are real rays and we draw them as solid lines.

We extend the three lines backward as dashed lines and note that

all three meet in a single point Q in front of the lens. To an

observer behind the lens, the light coming from O will seem to

come from Q and an upright, magnified image of the object O will

be seen. This image is a virtual image and not a real image since

no light actually issues from Q.

Figure 8 Magnifying glass.

By an appropriate choice of notation convention, we can apply the

thin lens formulae to the magnifying glass. By way of a specific

example, setting o = f / 2 in these equations, for instance, yields

i = – f, I = – 2O and M = -2. We interpret the minus sign in the

first equation as meaning that i extends now in front of the lens and

the minus sign in the second that the image is no longer inverted

but upright. We therefore introduce the following convention: o

and i are taken to be positive if the object is to one side and the

image on the other side of the lens. O and I are taken to be

positive if the object is upright and the image is inverted.

Figure 9 Biconcave lens.

-I



We can carry this one step further. Concave lenses (lenses that are

thinner in the center than on the rim) make parallel incident light

diverge. We formally assign to them a negative focal length.

Figure 91 shows that this is justified. To an observer behind such a

lens, the incident parallel rays do seem to have come from a virtual

focal point in front of the lens.

In this investigation, you will familiarize yourself with a

converging lens. You will first find the focal length of the lens and

then observe how such a lens creates an image.

For this investigation, you will need the following materials:

• Planoconvex lens made of Lucite

• Light ray box with five ray pattern

• 40 cm of graph paper from roll for focal length activity

• 60 cm of graph paper from roll for ray tracing

Activity 2-1: Finding the Focal Length

1. Place the ray box on top of a (new) piece of graph paper. Select

the five ray pattern by replacing the end piece.

2. In order to do this activity effectively the rays must enter the

lens parallel to one another. To adjust the rays, slowly move the

top of the box until the rays are parallel with the lines on the

graph paper.

F

f

Figure 10 Focal point of a planoconvex lens.

3. Place the lens in the center of your graph paper. Let the center

ray from the ray box pass through the center of the lens at a 90º

angle, as shown in Figure 10. Trace the position of the lens on

the graph paper and label the diagram.

1The lens shown in Figure 9 is a biconcave lens; Equations (16) and (17) apply to it as

well, as long as it is thin.



4. Note that the rays converge at a point on the other side of the

lens. This is the focal point F for the lens. To measure it, make

points that will allow you to trace the rays entering and leaving

the lens.

5. Remove both the lens and the ray box to measure f.

f cm

Prediction 2-1: What will happen if you place the lens backwards

over the position in steps 3 and 4? What will happen to the focal

point F and focal length f ?

6. Turn the lens around and place it at the previous position to

determine if the orientation of the lens influences its focal

length.

Question 2-1: Do the lines converge at the same point as the value

that you found in step 4? Should light incident on either side

collect at the same point? What does this tell you about the lens?

Activity 2-2: Ray Tracing

This activity is designed to test the imaging properties of the lens.

A ray-tracing diagram like the one shown in Figure 7 will be

created.

1. Place a clean 60 cm long piece of graph paper on the table.

2. Align the planoconvex lens somewhere on the graph paper.

Allow about 25 cm clear on either side of the lens. Draw the

central axis (see Figure 7). Draw around the lens to mark its

position and mark the two focal points F on the central axis on

either side of the lens. Use the value you found in Activity 2-1.

3. To test how an image is formed, you will draw an object arrow

like that shown in Figure 7 on your piece of graph paper. Place



the tip of the arrow at a distance of 2f from the lens and about

1.5 cm from the central axis. Record your values.

Object Distance (o): __________

Object Size (O): __________

Prediction 2-1: Using the focal length f , object size O and object

distance o that you measured above, use the thin lens formulae

[Equations (16) and (17)] to calculate the theoretical values for the

image distance and size and the magnification. Insert your

calculated values in Table 2-1.

Table 2-1

Image

distance

(i)

Image size

(I)

Magnification

Theory

Experimental

4. Using a single ray from the ray box, mark on your paper the ray

paths on both sides of the lens the rays shown in Figure 7. Use

a different marking scheme (e.g., •, ×, o) for points along each

of the three rays. Mark two points on either side of the lens to

help you draw the rays later after you remove the lens. Your

three rays should be as follows:

• Ray 1 should go through F on its way to the image point

• Ray 2 should enter the lens parallel to the optical axis

• Ray 3 should pass through the lens nearly unbent

5. Note where the three rays seem to indicate the image should be.

You have found the image of only one point – the tip of the

object arrow, but that is enough to deduce the entire image.

Draw an arrow indicating where the image is. Measure the

image distance and image size and fill in the experimental

values in Table 2-1. Calculate the magnification and enter it

into Table 2-1. Include your labeled diagram with your group

report.




experimental and theoretical values. [Do not be disappointed if

things do not work out exactly. Our “thin lens” is a bit on the

pudgy side.]

INVESTIGATION 3: IMAGE FORMATION BY CONVERGING LENSES

In order to examine the image formed by a converging lens, you

will need the following:

Optical bench Lens holder

100 mm lens2 200 mm lens

Illuminated object Viewing screen

Small see-through ruler 3 meter tape

Small desk lamp

Activity 3-1: Image Formation by a Converging Lens

In this activity, you will see the relative positions for the object and

image distances formed by a converging lens.

Image

Lens

Object

Figure 11 Creation of an inverted real image on the optical bench.

2 Lenses are labeled by focal length, not by any geometrical parameters such as a radius

of curvature.



Prediction 3-1: If the object is always outside of the focal point,

do you expect the image distance to increase or decrease if the

object distance is increased?

Prediction 3-2: What do you expect will happen to the image size

if the object distance is increased?

1. Place an illuminated object together with the mounted 100 mm

lens and the viewing screen on the optical bench as shown in

Figure 11.

2. Measure the size of the object, using the small ruler.

Object size: ____________ cm

3. Set the initial object distance to 15 cm.

4. Find the location of the image. To do this, move the screen

until a sharp image is formed. Record the image distance, as

well as the image size in the second two columns of Table 3-1.

Table 3-1 Experimental Data

Object

Distance

Image

Distance (cm)

Image Size

(cm)

Magnification Upright or

Inverted?

Image: Real

or Virtual?

15 cm

20 cm

30 cm

5. Calculate the magnification of your image and record in

Table 3-1.

6. Is the image upright or inverted? Real or virtual? Record your

observation in Table 3-1.



7. Try two other object distances, 20 cm and 30 cm. Record the

image distance, image size, magnification, orientation and

image properties of the image in Table 3-1.

Table 3-2 Theoretical Results

8. Use the thin lens formulae to calculate the image distance,

image size, and magnification for the three object distances

shown in Table 3-2. Let each lab partner calculate one. Enter

your calculated values into the table.

Question 3-1: How good is the agreement between your

experimental data in Table 3-1 and your calculations in Table 3-2?

Compare with your Predictions 3-1 and 3-2.

Prediction 3-3: All of the previous measurements involved an

object distance greater than the focal length. What will the image

look like if the object distance is less than the focal length?

9. Make sure that the object is oriented so it is facing the center of

the room and at the end of the optical bench furthest away from

the end of the table. This will make your upcoming

observations significantly easier.

10. Place the 10 cm lens so that the object distance is

approximately 5-8 cm.

Object distance Image distance

(cm)

Image size (cm) Magnification

15 cm

20 cm

30 cm



11. Stand at the end of the table so you are looking through the

lens at the object. Your distance to the lens should now be

approximately 1 m.

Question 3-2: Describe your image. Is it upright or inverted? If

you were to put a screen where you are looking, would an image

form there? What does this tell you about the image? Is it real or

virtual?

12. Now have one of the students in the group slowly move the

lens away from the object until it is approximately 10 cm from

the object. Make sure that another student is standing at the

end of the optical bench still looking through the lens.

Continue looking until the image disappears.

Question 3-3: Why is it that when the object is at the focal length

it produces no image? [Hint: consider the thin lens formulae.]

Activity 3-2: Test Fixed Distance

1. Position the lighted object 50 cm away from a viewing screen.

There will be two positions of the 100 mm lens where an image

will form. Let position “1” be where the lens is closest to the

object. [In your pre-lab you were asked find these distances

and to calculate the magnification for each position and these

results should be already entered into Table 3-3.]



Table 3-3

Object

distance (cm)

Image

distance (cm)

Image size

(cm)

Magnification

Pre-lab 1

2

Experiment 1

2

2. Move the lens until you find the two positions that produce

sharp images. Measure and record (in Table 3-3) the image and

object distances and image size for each position.

3. Calculate the magnification for both positions and enter your

results into Table 3-3.

Question 3-4: Discuss the agreement between your predictions

and your experimental results.

Activity 3-3: Simulating a Camera

1. Place the object at one end of the optical bench and the viewing

screen at the other end.

2. Place the 100 mm lens near the viewing screen and move the

lens until you see a focused image on the screen. (On a real

camera, a focus knob will move the lens elements toward or

away from the film.) Note the size of the image.

3. Repeat with the 200 mm lens . Is the image larger?



Question 3-5: Based on these results, would you expect a

telephoto3 lens to be shorter, longer, or the same length as a

“normal” lens? Explain.

Activity 3-4: A Telescope

In this activity, you will see how converging lenses are used in the

formation of telescopes.

1. This setup should be somewhere in the lab. You do not need to

create it on your optical bench.

2. The 100 mm lens (the eyepiece or ocular) and the 200 mm lens

(the objective) should be approximately 30 cm apart on the

optical bench.

3. Look through the 100 mm lens (toward the 200 mm lens). You

can adjust the distance between the lenses until objects across

the room are in sharp focus.

Question 3-6: Describe the image you see. Is it upright or

inverted? Magnified?

3 A “telephoto” lens makes distant objects look larger.



L11-1


Name Date Partners

Lab 11 - POLARIZATION

OBJECTIVES

• To study the general phenomena of electromagnetic wave

polarization

• To investigate linearly polarized microwaves

• To investigate linearly polarized visible light

OVERVIEW OF POLARIZED ELECTROMAGNETIC WAVES

Electromagnetic waves are

time varying electric and

magnetic fields that are

coupled to each other and that

travel through empty space or

through insulating materials.

The spectrum of

electromagnetic waves spans an

immense range of frequencies,

from near zero to more than

1030

Hz. For periodic

electromagnetic waves the

frequency and the wavelength

are related through

c fλ= , (1)

where λ is the wavelength of the

wave, f is its frequency, and c is

the velocity of light. A section

of the electromagnetic spectrum

is shown in Figure 1.

In Investigation 1, we will use waves having a frequency of

1.05 × 1010

Hz (10.5 GHz), corresponding to a wavelength of

2.85 cm. This relegates them to the so-called microwave part of the

spectrum. In Investigation 2, we will be using visible light, which

has wavelengths of 400 – 700 nm (1 nm = 10-9

m), corresponding to

frequencies on the order of 4.3 × 1014

-7.5 × 1014

Hz (430 - 750 THz).

These wavelengths (and hence, frequencies) differ by nearly five

orders of magnitude, and yet we shall find that both waves exhibit

the effects of polarization.

102

104

103

10

106

108

107

105

1010

1012

1011

109

1014

1016

1015

1013

1018

1020

1019

1017

1022 1023

1021

10-2

10-4

10-3

1

10-6

10-8

10-7

10-5

10-10

10-12

10-11

10-9

10-14 10-15

10-13

10-1

102

104 103

106

107

105

1 MHz

1 kHz

VISIBLE LIGHT

Frequency, Hz Wavelength, m

Gamma rays

X rays

Ultraviolet light

Infrared light

Short radio waves

Television and FM radio

AM radio

Long radio waves

1 nm

1 mm 1 cm

1 m

1 km

1 Å

Microwaves

1 GHz

1 THz

1 µm

10

Figure 1

L11-2 Lab 11 - Polarization


Electromagnetic waves are transverse. In other words, the

directions of their electric and magnetic fields are perpendicular to

the direction in which the wave travels. In addition, the electric

and magnetic fields are perpendicular to each other.

When the electric field of a wave is oriented in a particular direction,

that is to say, not in random directions, we say the wave is polarized.

In this workshop, we will investigate the polarization of two types of

electromagnetic waves that have somewhat different wavelengths

and frequencies: microwaves and visible light. We will both

produce and analyze polarized waves.

Figure 2 shows a periodic electromagnetic wave traveling in the

z-direction and polarized in the x-direction. E is the vector of the

electric field and B is the vector of the magnetic field. Study this

figure carefully. We will refer to it often.

Direction of

Propagation

Figure 2

Electromagnetic waves are produced whenever electric charges are

accelerated. This makes it possible to produce electromagnetic

waves by letting an alternating current flow through a wire, an

antenna. The frequency of the waves created in this way equals the

frequency of the alternating current. The light emitted by an

incandescent light bulb is caused by thermal motion that accelerates

the electrons in the hot filament sufficiently to produce visible light.

Such thermal electromagnetic wave sources emit a continuum of

wavelengths. The sources that we will use today (a microwave

generator and a laser), however, are designed to emit a single

wavelength.

The inverse effect also happens: if an electromagnetic wave strikes a

conductor, its oscillating electric field induces an oscillating electric

current of the same frequency in the conductor. This is how the

receiving antennas of a radios or television sets work. The associated

oscillating magnetic field will also induce currents, but, at the

frequencies we will be exploring, this effect is swamped by that of

the electric field and so we can safely neglect it.

Lab 11 - Polarization L11-3


Even though the electric field vector is constrained to be

perpendicular to the direction of propagation, there are still

infinitely many orientations possible (illustrated in Figure 3).

Electromagnetic waves from ordinary sources (the sun, a light

bulb, a candle, etc.), in addition to having a continuous spectrum,

are a mixture of waves with all these possible directions of

polarization and, therefore, don’t exhibit polarization effects.

Some possible directions of

the electric field vector

Direction of

propagation

Figure 3

It is, however, possible to produce linearly polarized

electromagnetic waves. In other words, waves whose electric

field vector only oscillates in one direction. Look again at

Figure 2. It schematically shows a linearly polarized

electromagnetic wave polarized in the x-direction.

The electric field of a plane wave of wavelength λ, propagating in

the z-direction and polarized in the x-direction, can be described

by:

−= )(

2sin ctzExx

λ

πiE , (2)

where Ex is the vector of the electric field, Ex its amplitude, and i

the unit vector in the x-direction. A wave of the same wavelength,

polarized in the y -direction, is described by:

+−= φ

λ

π)(

2sin ctzEyy jE . (3)

Here, j is the unit vector in the y-direction and φ is a constant that

accounts for the possibility that the two waves might not have the

same phase. From two such waves, one can construct all plane

waves of wavelength λ traveling in the z-direction.



y

x

E

a) b) c)

E E

x

y y

x θ

Figure 4

If both x- and y-components are present and their phase difference

is zero (or 180°), the wave will be linearly polarized in a direction

somewhere between the x -direction and the y -direction, depending

on the relative magnitudes of Ex and Ey (see Figure 4a).

Mathematically such a wave is described by:

−±=+= )(

2sin)( ctzEE yxyx

λ

πjiEEE , (4)

where the plus sign refers to a phase difference of zero and the

minus sign to one of 180° (π radians). The angle θ between this

polarization direction and the x -direction is given by

x

y

E

E=θtan . (5)

If the phase shift is not zero (or 180°), the wave will not be linearly

polarized. While we will only be investigating linear polarization

in this lab, it is useful to know something about other types of

polarization. Consider the case where the magnitudes are equal,

but the phase shift is ±90° (± π/2 radians). In other words:

yx EE = and 2

πφ = ± , (6)

The resulting wave, called a circularly polarized wave, can be

written:

2 2

sin ( ) cos ( )x y E z ct z ctπ π

λ λ

= + = − ± −

E E E i j (7)

by making use of the fact that απα cos)2/(sin ±=+ . With the

plus sign, this equation describes a wave whose electric field vector,

E, rotates clockwise in the x -y plane if the wave is coming toward

the observer. Such a wave, illustrated by Figure 4b, is called a right

circularly polarized wave. With the minus sign, the equation

describes a left circularly polarized wave.



With the phase shift still ±90°, but with different magnitudes

2

EE yx

πφ ±=≠ and , (8)

the E vector will still rotate clockwise or counterclockwise but will

trace out an ellipse as shown Figure 4c.

With thermal sources, there is a random mix of different Ex, Ey,

and φ values. The resulting wave will be unpolarized.

Polarized electromagnetic waves can be obtained in two ways:

1. by using sources, such as certain lasers, that produce only waves

with one plane of polarization, or

2. by polarizing unpolarized waves by passing them through a

polarizer, a device that will let only waves of one particular

plane of polarization pass through.

Some sources of electromagnetic waves generate linearly polarized

waves. Examples include the microwave generator we'll use today

as well as some types of lasers. Other sources generate unpolarized

waves. Examples include thermal sources such as the sun and

incandescent lamps.

One way of producing linearly polarized electromagnetic waves

from unpolarized sources is to make use of a process that directs

waves of a given polarization in a different direction than waves

polarized in the perpendicular direction. Earlier we noted that the

electric field of an electromagnetic wave incident upon a wire

induces an oscillating current in the wire. Some energy will be lost

through resistive heating, but most will be re-radiated (scattered).

Only the component of the oscillating electric field that is parallel

to the wire will induce a current and be scattered. The electric

field component perpendicular to the wires, on the other hand, will

be essentially unaffected by the wires (assuming a negligible wire

diameter). Hence, both the scattered and unscattered

electromagnetic waves are linearly polarized.

For microwaves, we can (and will) use an array of actual wires.

For visible light, we use a Polaroid filter. Polaroid filters are

made by absorbing iodine (a conductive material) into stretched

sheets of polyvinyl alcohol (a plastic material), creating, in effect,

an oriented assembly of microscopic “wires”. In a Polaroid filter

the component polarized parallel to the direction of stretching is

absorbed over 100 times more strongly than the perpendicular

component. The light emerging from such a filter is better than

99% linearly polarized.



Polarizer P

θ

Ei

Ee

p

Figure 5

A polarizer will only pass the components of an electromagnetic

wave that are parallel to its polarizing axis. Figure 5 shows

polarized electromagnetic waves incident on a polarizing filter, P

(shown as a wire array).

The electric field of the incident wave (Ei) is oriented at an angle θ

relative to the polarization axes of P. Let p be a unit vector along

the polarization axis of the polarizer. The effect of the polarizer,

then, is to “project out” the component of Ei that is along p:

Ee = p (p·Ei) = Ei cos θ p. Because the intensity of an

electromagnetic wave is proportional to the square of its electric

field amplitude, it follows that the intensity of the electromagnetic

waves exiting the analyzer is given by:

2cose i

I I θ= . (9)

This is known as Malus’ Law, after the French physicist who

discovered the polarizability of light.

Initially unpolarized electromagnetic waves can be thought of as a

mixture of all possible polarizations. Each possible polarization

will be attenuated according to Malus’ law, and so the total

intensity will be the initial intensity times the average of 2cos θ

(which is 1/2). In other words, the intensity is reduced to one half

of the incident intensity.

Except in the case where θ is zero (or 180°), Ee (the electric field

of the electromagnetic waves exiting the polarizer) will have a

component that is perpendicular to Ei. If we place yet another

polarizer after P (call it P') with its polarization axis right angles to

incident wave’s polarization axis, we will get electromagnetic

waves out whose polarization is orthogonal to the incident waves’

polarization. We have effectively rotated the polarization of the

incident waves (with some loss of intensity). Applying Malus’

Law, we get:

2 2cos cosi

I I θ θ′ ′= (10)

where θ is the angle between the initial polarization and the first

polarizer, P, and θ' is the angle between P and the second polarizer,



P'. But P' is at right angles to the initial wave's polarization, so

90θ θ ′+ = ° . Hence, 2 2cos sinθ θ′ = . Using another trigonometric

identity (sin 2θ = 2 sin θ cosθ), we finally get 20.25 sin 2i

I I θ′ = .

We can see we get the maximum transmission when θ = 45°

(sin 2×45° = 1) and that it is one quarter of the intensity of the incident polarized waves (Ii).

INVESTIGATION 1: MICROWAVE POLARIZATION

For this experiment, you will need the following:

• Gunn diode microwave transmitter

• Microwave receiver

• Wire grid polarizer

CAUTION: DO NOT ALLOW THE RECEIVER’S METER

TO PEG AT ANY TIME!

To peg the meter means to allow the needle to go beyond the

maximum value on the scale. If you find the meter pegged, immediately turn down the sensitivity and/or move the receiver

away from the microwave generator!

Activity 1-1: Polarization of Microwaves from a Gunn Diode

Inside the microwave generator is a solid state device called a Gunn diode. When a DC voltage is applied to a Gunn diode,

current flows through it in bursts at regular intervals. For your diode, these bursts come at 9.52 × 10

-11 seconds apart causing, in

addition to the dc current, an ac current at 1.05 × 1010

Hz (10.5 GHz). As a result, a large AC voltage, oscillating at that

frequency, is present across the slot, and so a wave is radiated from the horn. The electric field of this wave oscillates in the same

orientation as the Gunn diode. The polarization of an electromagnetic wave is determined by the direction of the electric vector E. The magnetic field B encircles the current in the Gunn

diode and so emanates in the orientation perpendicular to E.

Important Note: The Gunn diode is place inside the generator in a

way that the electric field will oscillate vertically when the knob on

the back is placed at 0º.

Just inside the horn of the receiver is a microwave detector. In

addition, there is some circuitry, which amplifies the signals received by the detector and outputs this amplified signal to a

d’Arsonval meter and to an external output. The sensitivity (labeled METER MULTIPLIER) is controlled via two knobs. The

VARIABLE SENSITIVITY knob allows for fine adjustment. As



you turn up the sensitivity (from 30 to 1), the signal is amplified more and more.

Generator

Receiver

Figure 6

1. Set up the generator and receiver as shown in Figure 6, with about 75 cm between the faces of the horns.

Prediction 1-1: With what relative orientation of the transmitter and receiver do you expect to find minimum intensity? What does

this tell you about the electromagnetic microwaves?

Set the knobs on back of both pieces so the angle indicator is at 0°. Adjust the sensitivity on the receiver to obtain a signal near 0.5 on

the meter. If you cannot achieve this with a sensitivity of 10 or 3, move the receiver closer to the generator. Rotate the receiver and

verify that it is sensitive to the polarization of the wave. Return the receiver angle to 0º.

Question 1-1: Does it make sense that maximum intensity is obtained when both generator and receiver are oriented the same

way? Explain why. Why does the received signal go to zero when they are at 90º with respect to one another?



Activity 1-2: Wire Grid Polarizer

Prediction 1-2: With the generator and receiver oriented the same

way, what orientation (relative to the generator) of a wire grid placed in between them will give the maximum received intensity?

Prediction 1-3: With the generator and receiver oriented at 90°

with respect to one another, what orientation (relative to the generator) of a wire grid placed in between them will give the

maximum received intensity?

1. Make sure that the generator and the receiver are oriented the same way: with the E field horizontal (indicators at ±90°).

2. Insert the wire grid polarizer between the generator and the receiver so that the wires are initially oriented horizontally

(parallel to the direction of the E field). Slowly rotate the polarizer so that the wires become perpendicular to the E field.

Question 1-2: With what relative orientation(s) of the polarizer did the receiver indicate the highest intensity? The lowest?

3. Rotate the receiver’s angle by 90º so that the generator and

receiver are orthogonal and turn up the sensitivity to 1.

4. Repeat step 2.



Question 1-3: With what orientation(s) of the polarizer did the receiver indicate the highest intensity? The lowest?

NOTE: Turn off your receiver and unplug the generator.

INVESTIGATION 2: POLARIZATION OF A HIGH-INTENSITY LAMP

NOTE: IN THE REMAINDER OF THE WORKSHOP WE

WILL INVESTIGATE THE POLARIZATION OF VISIBLE

LIGHT. FOR THE NEXT THREE INVESTIGATIONS, IT

WILL BE NECESSARY TO TURN OFF ALL OF THE

LIGHTS IN THE LAB TO OBTAIN THE BEST RESULTS.

In this investigation, the unpolarized light from a high-intensity lamp will be linearly polarized. This polarization will be

investigated with a second Polaroid analyzer. In addition, a third polarizer will be added to investigate the effect of the orientation

of a third polarizer on the intensity.

For this you will need the following:

• Optical bench with lens holders

• Polarizers

• Polarized light demonstrator kit

• Goniometer

• Small support stand

• Desk lamp (high intensity light source)

• Light sensor and cable

Activity 2-1: Linearly Polarized Light and Malus’ Law

Note: The light sensor that will be used for the rest of the experiments is a photodiode with a sensitivity that ranges from 320 nm to 1,100 nm. Make sure not to allow the output voltage

from the sensor to go above 4.75 volts. At this point, the sensor

is saturated and you will not get accurate readings.



Figure 7

1. Set up the lamp, polarizers, and light sensor as shown in Figure 7. DO NOT TURN YOUR LAMP ON YET. Make

sure your lamp is on the opposite end of the table from the computer and is pointing towards the wall, not towards the

center of the room. We want to minimize the interference of the light coming from the desk lamp into each other’s light

sensor.

2. The heat-absorbing filter (item #1 in the box of components)

should be mounted on the small support stand in between the light source and the first polarizer. Place it as close to the

polarizer as you can so that little, if any, light can get into the polarizer without first passing through the heat filter.

3. Ensure that the heights of the light, heat absorbing filter, polarizers and light sensor are lined up. Your lamp can now be

turned on.

4. Look through the analyzer at the lamp. Play around with the

relative orientation of the two polarizers. Record your observations.

ALWAYS PLACE THE HEAT ABSORBING FILTER

BETWEEN THE LIGHT AND THE FIRST POLAROID

FILTER TO BLOCK THE INFRARED LIGHT AND

PREVENT HEAT DAMAGE

Note: The infrared light emitted from the lamp will not be

polarized by the filters, but will be seen by the photodetector.



5. Connect the light sensor to channel A in the PASCO interface.

6. Open the experiment file named L11A2-1 Linearly Polarized.

There should be a data table when you open the file.

7. In the data table, the first column will be the values for angle

that you enter. The units for intensity are not volts but Lumens. However, the output of the light sensor probe in volts

is directly proportional to the light intensity. Never let the

output from the light sensor exceed 4.75 V

8. Ensure that the light from the lamp is incident on the heat absorbing filter, then travels through the first polarizer, then

through the analyzer, and then onto the light sensor.

9. Make sure that the two polarizers are aligned (both at 0º).

10. Press Start to begin collecting data. The output from the light sensor will be shown in the digits window and in the third

column of the first row. A typical voltage reading when the lamp and sensor are 50-60 cm apart is in the range of 0.5 to

1 V. There is a sensitivity switch on the light sensor that you may need to adjust.

11. Set the first polarizer to 0º and the analyzer to –90º (counterclockwise value; see Figure 8 for one possible

convention). The Polaroid Filters we are using allow the

electric field E vector of the transmitted light to oscillate in the

direction of the indicator tab on the Polaroid.

12. When you feel that the

reading has stabilized, press Keep. A box will pop up that

asks you the angle of the polarizer. Type in “-90” and

press Enter.

13. Adjust the analyzer to an

angle of -80º. The voltage output will now be shown as

before. Press Keep and type in the angle.

14. Adjust the angle of the analyzer in 10º steps from

-90º to 90º. Repeat step 12 until all of the values are entered, putting in the respective

values for the angles. This can go rather quickly with one person changing the angle and another person operating the

computer. Once Keep has been selected, the next angle can be changed by one group member while another is entering the

angle into the computer.

Figure 8



15. When you are finished entering data, click on the red square next to Keep to stop data collection.

16. Print out your table for your report. Only print one per group.

17. At the bottom of the screen, there should two graphs

minimized. Bring up the graph titled I vs. Angle so you can see the graph of your light intensity plotted versus angle. If

you see a fit to your data, you have brought up the wrong graph.

Question 2-1: What does your graph look like? Does it follow the curve you would expect?

18. Minimize this graph, and maximize the second graph entitled

Fit Malus. You will see your data plotted along with a fit. You could have easily entered this fit into Data Studio

yourself, but we have done it for you to save time. We have fit straight line (y = mx + b) to I versus cos

2θ.

19. Record the fit parameters m and b:

m b

Question 2-2: Discuss the physical meanings of m and b.

Question 2-3: Is it possible that b is not constant? Explain. How could you minimize b?



20. Print out your Fit Malus graph and attach it at the end of your lab. Only print one per group.

Note: The following experiment will use all of the setup from

Activity 2-1. Leave everything in place.

Activity 2-2: Three Polarizer Experiment

1. Using the setup from Activity 2-1, set the two existing polarizers so that they are crossed (e.g. the polarizer at 90º and

the analyzer at 0º).

Make sure to leave the heat absorbing filter in place between

the lamp and the polarizers!

Prediction 2-1: What is the orientation of the electric field after it passes through the first polarizer? What will happen to this light

when it reaches the second polarizer?

Prediction 2-2: With this arrangement, what output do you expect from the light sensor?

Prediction 2-3: A third polarizer will be added in between the other two. What effect will this have, if any, on the output of the

light sensor?



Prediction 2-4: What orientation of the third polarizer (in between the first two) do you expect would produce maximum

voltage? Give the angle with respect to the first polarizer.

2. Place the third polarizer in between the other two.

3. Look through the analyzer at the lamp. Play around with the relative orientation of the middle polarizer. Record your

observations.

4. Open the experiment file named L11A2-2 Three Polarizer. There should be two digit displays; one for voltage output and

one for intensity. Press Start to activate the displays.

5. Adjust only the middle polarizer and find the orientation for

which the output shown on the computer is a maximum. Record the angle at which the maximum occurs.

Angle

6. Click on the red square to stop the data collection.

Question 2-5: Explain your findings in terms of the orientation of the electric field after the light travels through each polarizer. Why

would the angle found in step 5 produce the maximum intensity?



INVESTIGATION 3: BREWSTER’S LAW

An alternative way to produce linearly polarized light is based on Brewster’s law. A wave falling on the interface between two

transparent media is, in general, partly transmitted and partly reflected. However, there is a special case in which the directions

of the refracted and reflected waves are perpendicular to each other, as shown in Figure 9.

α α

n1

Incident

Ray Reflected

Ray

n2

β Refracted

Ray

Figure 9

The component of the wave whose electric field vector E is in the

plane of the page, called the p wave, is not reflected at all but completely transmitted when the incident angle is α (called

Brewster’s angle) from the normal. The electric field of the p wave is represented by the short lines () in the figure.

Meanwhile, the reflected light contains the remainder of the wave, the component whose electric vector oscillates perpendicular to

the plane of the page. Therefore, the light that is reflected is totally polarized. This second wave is usually called the s wave.

The electric field of the s wave is represented by the dots () in the figure.



The angle of incidence satisfying the condition of Brewster’s law, called Brewster’s angle, is easily obtained from Figure 9. Noting

that

2

πα β+ = (11)

and using Snell’s law ( 1 2sin sin ,n nα β= where n1 is the index of

refraction of the medium containing the incident ray, and n2 is the index of refraction of the medium containing the refracted ray), we

can show:

2

1

sin sin sintan

sin cossin

2

n

n-

α α αα

πβ αα

= = = =

. (12)

In the case that you will be looking at in class, the index of refraction of the first medium, n1 is equal to the index of refraction

of air. For this workshop, this will be taken to be unity. Putting this into Equation (12), we get:

tann α= (13)

where n is the index of refraction of the glass plate.

Brewster’s law is just a special case of the Fresnel equations that

give the amplitudes of the transmitted and reflected waves for all angles for the two polarizations.

The polarization upon reflection is rarely used to produce polarized light since only a few percent of the incident light are reflected by

transparent surfaces and become polarized (metal surfaces do not polarize light on reflection). But the fact that light reflected by

glass, water, or plastic surfaces is largely polarized enables one to cut down glare with Polaroid glasses or Polaroid photographic

filters.

α

α

β

β

n

Figure 10



If one shines light at the Brewster angle onto a plane parallel glass plate, as shown in Figure 10, the Brewster condition is satisfied at

both the entrance and the exit face. This means that the p wave is perfectly transmitted (without reflection) by both surfaces. Such

an arrangement is called a Brewster window. Such windows are often used in gas lasers. As a result, the light from these lasers is

strongly linearly polarized.

In this investigation, a laser will be used to test Brewster’s Law. For this investigation, you will need the following:

• Laser

• Polarizer

• Glass plate taped to mount

• Goniometer [Basically two sticks pinned together and a protractor to measure the angle between them. Today we won’t be using the sticks, just the protractor.]

• Small support stand

WARNING: LASER LIGHT CAN DAMAGE THE EYES.

NEVER LOOK DIRECTLY INTO THE BEAM OR AT

LASER LIGHT REFLECTED FROM METAL, GLASS OR

POLISHED SURFACES.

Activity 3-1: Determination of Brewster’s Angle

Prediction 3-1: You will be using crown glass as your Brewster

window in the following experiment. What angle do you expect to find, knowing the index of refraction of crown glass (see

Appendix A)?

1. Place the goniometer on the table next to the middle of the optical bench. Place the mounted glass plate on the goniometer

hinge. This will serve as your Brewster window. See Figure 11.

Figure 11



2. Adjust the laser so that the beam produced is horizontal and is incident upon the glass plate. The distance between the laser and

glass plate should be about 50 cm, but this does not require a measurement.

3. Place a polarizer and holder on the table in between the laser and the glass plate, so the light travels through the polarizer.

Recall that only the s wave (electric field vector E parallel to the glass plate) is reflected at Brewster's angle (see Figure 9). If the

s wave is not present in the incident light, then the Brewster’s angle can be found quite easily; it will be the point at which no

light is reflected. We want to use the polarizer to only allow the p wave (electric field vector E in horizontal plane) to be incident

upon the glass plate.

Question 3-1: At what angle should we set the polarizer to

transmit only the p wave?

Polarizer angle for only p wave transmission:

Explain how you decided upon this angle:

Note: Make sure that the polarizer does not completely block the laser light. To check this, look at the glass plate to ensure that there is light incident upon it. Also try to find the refracted beam. No

matter what the angle of the glass plate is with respect to the beam, there will always be a refracted (transmitted) beam – the p wave is

always refracted.

4. Set the polarizer at the angle you just determined in

Question 3-1.

5. You may find that your laser is polarized (some of our lasers

are, some aren’t). If so, you may have to rotate the laser about its axis so that enough of the beam passes through the polarizer

to clearly visible.

6. Let the reflected light fall upon a notebook size piece of white

paper as a screen to show the reflected ray, rotate the glass plate until you find the position at which the intensity of the reflected

light becomes a minimum. You may find that by alternately



“tweaking” the polarizer angle and the plate angle, you can make the reflected ray completely vanish.

Note: Be careful to position the laser beam so that it does not miss the glass plate as you rotate the spectrometer stand. Do not let the

laser beam shine into the computer screen or the light sensor.

7. Read the angle on the goniometer for the position for which the light is a minimum.

θ

8. Rotate the glass plate until the beam is reflected back into the

laser. Read the angle on the goniometer again and consider this your zero angle.

0θ

9. Find the value for αθθ =− 0 . This is your Brewster’s Angle.

α

Question 3-2: Does your experimental value of the Brewster’s

angle agree with your Prediction 3-1? If not, explain.

INVESTIGATION 4: OTHER METHODS FOR POLARIZING AND

DEPOLARIZING

DEPOLARIZATION

To change polarized light into unpolarized light one must

introduce random phase differences between the two components of the electric vector. This can be accomplished by interposing a

material that is both inhomogeneous and anisotropic across the wave front.

BIREFRINGENCE

Most of the transparent materials that one encounters daily, such as

glass, plastics, and even crystalline materials such as table salt, are optically isotropic, i.e. their index of refraction is the same in all

directions.

Some materials, however, have an optically favored direction. In

these materials the index of refraction depends on the relative



orientation of the plane of polarization to that preferred direction. Such materials are called birefringent or doubly refracting.

The best known example of a birefringent material is calcite (CaCO3). Normally optically isotropic materials, such as glass,

can be given a preferred direction (and thus made to be birefringent) by stressing or bending them.

1

2

o-ray

e-ray

Figure 12

Consider a light wave traversing a birefringent crystal, as shown in

Figure 12, where the direction of propagation of the wave is

entering the crystal perpendicularly. An initially unpolarized light

beam will split into two separate linearly polarized beams. One of

these is called the ordinary ray or o-ray and the other the

extraordinary ray or e-ray. The behavior of the o-ray is

essentially that of a ray in an isotropic medium: it is refracted in

accordance with Snell’s law, and its refractive index no is

independent of the direction of travel.

The e-ray, on the other hand, behaves in a most peculiar way. Its

index of refraction ne depends on the orientation of the crystal.

Moreover, its direction of travel, after entering the crystal is not

consistent with Snell’s law. As Figure 12 shows, it will be

refracted even if its angle of incidence is 90°. On leaving the

crystal it becomes again parallel to the direction of incidence but

displaced with respect to the incident beam. Since the two

emerging rays are linearly polarized along mutually perpendicular

directions, doubly refracting crystals make very effective

polarizers: If one cuts a birefringent crystal so that the e-ray, but

not the o-ray, is totally reflected at the exit face one can produce

light that is 99.999% linearly polarized.

Another application of birefringence is the quarter-wave plate, a

device that can be used to convert linearly into circularly polarized

light and vice versa. Consider again Figure 12: Not only do the e-

and the o-rays have different speeds (due to the different indices of

refraction), but they also travel different distances in the crystal.

As a result they will be out of phase with respect to one another.

Through a suitable choice of the thickness of the crystal one can

arrange it that the phase difference between the two rays becomes



a quarter of a wavelength, in which case a linearly polarized

incident light beam will be circularly polarized on leaving the

crystal.

The wavelength dependence of the index of refraction, although

small, lends itself to some pretty demonstration experiments. If

one places two Polaroid filters in front of a light source so that

their directions of polarization are perpendicular to each other, they

will appear dark. If one then places an object made of a

birefringent material between the crossed Polaroids, a multicolored

image of the object will become visible in the previously dark

field. The o-ray and the e-ray have traveled different optical path

lengths and their phases, upon leaving the object, will differ, the

difference being a function of the wavelength of the light. Since

the two rays are polarized in different directions they cannot

interfere with each other. The second Polaroid (the analyzer)

passes that component of each ray whose plane of polarization is

parallel to the direction of polarization of the filter. These

components have the same plane of polarization and can interfere.

Whether their interference is constructive or destructive will

depend on their phase difference and hence on their color.

In this investigation, you will use different objects and materials to

both polarize and depolarize light. You will need the following

materials:

• Polarized light demonstrator kit

• Optical bench with polarizers

• Small support stand with tripod base and lens holder

• Desk lamp

Activity 4-1: Depolarization

As you will recall from the readings above, random phase

differences may be introduced between the two components of the

electric field vector to depolarize the light.

Incident

light

P P'

wax

paper

Figure 13



1. Set up two polarizers with the desk lamp and heat-absorbing

filter as done in Investigation 2 (see Figure 13). Set the

polarizer P such that E is vertical.

Prediction 4-1: With this setup (without wax paper), what do you

expect to see as you vary angle of the analyzer P' with respect to

P? (Hint: we did this in Investigation 2).

Prediction 4-2: With the wax paper added between the polarizers,

what do you expect to see as you vary angle of the analyzer P' with

respect to P?

2. Hold the piece of wax paper from the polarizing kit in between

the two polarizers.

3. Rotate the polarizer through 180º and observe (by eye) the

transmitted light.

Question 4-1: Describe the transmitted light intensity as you

rotate the polarizer.

Question 4-2: What does this show you about the polarization of

the light through the wax paper?



Activity 4-2: Birefringence by the Calcite Crystal

Set the calcite crystal from the polarization kit on the dot: •

1. Hold a polarizer over the calcite and look through it at the dot.

Question 4-3: What do you observe and with and without the

polarizer?

2. Slowly rotate the polarizer until only one dot is seen. Note the

orientation of the polarizer.

θ1 (choose a relative zero angle)

3. Rotate the polarizer again until the other dot is seen. Note the

orientation of the polarizer.

θ2 (use same zero angle as before)

Question 4-4: What does this tell you about the relative

polarization of the images created by the calcite crystal?

Activity 4-3: Interference Caused by Birefringence

1. Hold the mica sample between two crossed polarizers (set at 90º

and 0º, for example) and look through the setup at the lamp.

Incident

light mica plate

P P'

Figure 14



2. Tilt the mica sample slowly backwards as shown in Figure 14.

Question 4-5: What do you see?

Question 4-6: Do your observations depend on the angle at which

you hold the mica?

Activity 4-4: Birefringence Due to Stress

Replace the mica with the U-shaped piece of plastic between the

crossed polarizers.

1. Look through the polarizer at the plastic.

Question 4-7: What do you observe? Do you see light?

Question 4-8: Based on your previous observations, is the light

polarized by the plastic? Why or why not?

2. Lightly squeeze the two legs of the U toward each other while

looking at the plastic through the polarizer.



Question 4-9: What do you observe?

Question 4-10: What has changed about the light through the

plastic?

Comment: The strain partially orients the molecules and makes

the plastic birefringent. From such patterns engineers can locate

regions of high strain in a plastic model of a structure and then

decide whether the structure must be redesigned or strengthened in

certain places.

Question 4-11: In which corner of your plastic is there the greatest

stress?

Activity 4-5: Polarization of Scattered Light

Sunlight is scattered while passing through the atmosphere. Light

with a short wavelength is scattered more than light with a long

wavelength. This is why the sky appears blue. Light scattered by

90° is strongly polarized. You can verify this on a clear day if you

look through a Polaroid filter in the appropriate direction of the

sky.

A similar observation can be made in the laboratory by passing

laser light through a tank of water that has been clouded by

suspending some scattering material in it. At the front of the room

there should be such a tank with a laser beam should already be

passing through it.



1. From the side of the tank, at a right angle with respect to the

direction of the light, examine the scattered light using a

polarizer.

Question 4-12: Record your observations and use them to discuss

the polarization of the scattered light.



L12-1


Name Date Partners

Lab 12 - INTERFERENCE

OBJECTIVES

• To better understand the wave nature of light

• To study interference effects with electromagnetic waves in

microwave and visible wavelengths

OVERVIEW

Electromagnetic waves are

time varying electric and

magnetic fields that are

coupled to each other and

that travel through empty

space or through insulating

materials. The spectrum of

electromagnetic waves spans

an immense range of

frequencies, from near zero

to more than 1030

Hz. For

periodic electromagnetic

waves the frequency and the

wavelength are related

through

c fλ= (1)

where λ is the wavelength of

the wave, f is its frequency,

and c is the velocity of light.

A section of the

electromagnetic spectrum is

shown in Figure 1.

In Investigation 1, we will use waves having a frequency of

1.05 × 1010

Hz (10.5 GHz), corresponding to a wavelength of

2.85 cm. This relegates them to the so-called microwave part of the

spectrum. In Investigation 2, we will be using visible light, which

has wavelengths of 400 - 700 nm (1 nm = 10-9

m), corresponding to

frequencies on the order of 4.3 × 1014

- 7.5 × 1014

Hz

(430 - 750 THz). These wavelengths (and hence, frequencies) differ

102

104

103

10

106

108

107

105

1010

1012

1011

109

1014

1016

1015

1013

1018

1020

1019

1017

1022 1023

1021

10-2

10-4

10-3

1

10-6

10-8

10-7

10-5

10-10

10-12

10-11

10-9

10-14 10-15

10-13

10-1

102

104 103

106

107

105

1 MHz

1 kHz

VISIBLE LIGHT

Frequency, Hz Wavelength, m

Gamma rays

X rays

Ultraviolet light

Infrared light

Short radio waves

Television and FM radio

AM radio

Long radio waves

1 nm

1 mm

1 cm

1 m

1 km

1 Å

Microwaves

1 GHz

1 THz

1 µm

10

Figure 1

L12-2 Lab 12 - Interference


by nearly five orders of magnitude, and yet we shall find that both

waves exhibit the effects of interference.

Electromagnetic waves are transverse. In other words, the

directions of their electric and magnetic fields are perpendicular to

the direction in which the wave travels. In addition, the electric

and magnetic fields are perpendicular to each other.

Figure 2 shows a periodic electromagnetic wave traveling in the

z-direction and polarized in the x-direction. E is the vector of the

electric field and B is the vector of the magnetic field. Study this

figure carefully. We will refer to it often.

Direction of

Propagation

Figure 2

Electromagnetic waves are produced whenever electric charges are

accelerated. This makes it possible to produce electromagnetic

waves by letting an alternating current flow through a wire, an

antenna. The frequency of the waves created in this way equals the

frequency of the alternating current. The light emitted by an

incandescent light bulb is caused by thermal motion that accelerates

the electrons in the hot filament sufficiently to produce visible light.

Such thermal electromagnetic wave sources emit a continuum of

wavelengths. The sources that we will use today (a microwave

generator and a laser), however, are designed to emit a single

wavelength. Another essential characteristic of these two sources is

that they emit radiation of definite phase. That is to say, they are

coherent.

The inverse effect also happens: if an electromagnetic wave strikes a

conductor, its oscillating electric field induces an oscillating electric

current of the same frequency in the conductor. This is how the

receiving antennas of radios and television sets work. The associated

oscillating magnetic field will also induce currents, but, at the

frequencies we will be exploring, this effect is swamped by that of

the electric field and so we can safely neglect it.

Lab 12 - Interference L12-3


Electromagnetic waves carry energy. The energy density at any

point is proportional to the square of the net electric field. The

intensity (what we can observe) is the time average of the energy

density. Important Note: To find the intensity of the

electromagnetic waves at any point, we must first add up (as

vectors, of course), all of the electric fields to find the net electric

field. We cannot simply add intensities. It is this property of

electromagnetic waves1 that leads to interference effects.

In this workshop you will be studying how electromagnetic waves

interfere. We will, once again, be using two small regions of the

electromagnetic spectrum: microwaves and visible light. Look at

Figure 1 to understand the relative position of microwaves and

visible light. The microwaves that you will be using in this

experiment have a frequency of 1.05 × 1010

Hz, corresponding to a

wavelength of 2.85 cm. The name microwave is to be understood

historically: In the early days of radio the wavelengths in use were

of the order of hundreds, even thousands, of meters. Compared with

these waves, those in the centimeter region, which were first used in

radar equipment during World War. II, were indeed ‘micro’ waves.

You will recall that conductors cannot sustain a net electric field.

Any externally applied electric field will give rise to a force on the

free electrons that will cause them to move until they create a field

that precisely cancels the external field (thereby eliminating the

force on the electrons). If an electromagnetic wave strikes a

conductor, the component of its oscillating electric field that is

parallel to the wire will induce an oscillating electric current of the

same frequency in the conductor. This oscillating current is simply

the free electrons in the wire moving in response to the oscillating

external electric field.

Now you will also recall that an oscillating electric current will

produce electromagnetic waves. An important thing to note about

these induced waves is that their electric fields will be equal in

magnitude and opposite in direction to the incident wave at the

surface (and inside) of the conductor.

1 This is a general property of waves, not just for electromagnetic waves.



Figure 3

Consider now a plane electromagnetic wave incident upon a wire.

Figure 3 schematically shows a top view of such a case. Only one

incident wave front and the resultant induced wave front are shown.

The incident wave front is shown as having passed the wire and is

traveling from the top of the figure to the bottom. The induced wave

will be of the form of an expanding cylinder centered on the wire

and, since the induced wave travels at the same speed as does the

incident wave, the cylinder’s radius is equal to the distance that the

incident wave front has traveled since it struck the wire. At the point

where the induced wave and the incident wave touch, they add

destructively as the induced wave is 180° out of phase with respect to

the incident wave.

Figure 4

Figure 4 shows the same situation, but with a number of wires all

oriented in the same direction. We can see that the induced wave

fronts all line up in phase at the incident wave front. However, since

the induced waves are 180° out of phase with the incident wave, the

resulting wave front is reduced in amplitude. We indicate this by

Direction of Propagation

of Incident Wave Front

Incident Wave

Front

Induced

Wave

Front Wire

Direction of

Propagation of Incident

Wave Front



showing the incident wave front as a dashed line. With enough such

wires, the amplitude for the forward direction can be reduced to a

negligible level. Note that the energy of the incident wave is not lost;

it is simply re-radiated in other directions.

Figure 5

Figure 5 is similar to Figure 4 except that the wires are now arranged

in a linear array. We recognize this arrangement as the “wire grid

polarizer” from an earlier lab. In that earlier lab, we investigated the

polarization properties of the transmitted electromagnetic waves.

We now consider the properties of the scattered or reflected waves.

In Figure 5, we see that not only do the induced waves line up with

each other in the plane of the incident wave, now they also line up

with each other in another plane. This alignment of wave fronts

gives rise to constructive interference, meaning that the resulting

wave front’s amplitude is enhanced in this direction. With enough

wires, essentially all of the incident wave’s energy will be radiated in

this direction.

Furthermore, we can see from Figure 5 that the angle that this

reflected wave front makes with the plane of the wires is the same as

that of the incident wave front. In other words, “angle of incidence

equals angle of reflection”. We will find that, for microwaves, the

“wire grid polarizer” makes a fine mirror (but only for waves with

their electric fields aligned with the wires!).

What about Polaroid glasses and filters? Why do they not act like

mirrors? The answer is that Polaroid filters and glasses are thick

relative to the wavelength of visible light. The conductive molecules

are randomly distributed throughout the filter and, hence, are

arranged more like the wires shown in Figure 4 than in Figure 5.

Reflected

Wave

Front

Incident

Wave

Front


of Reflected Wave Front


of Incident Wave Front



INVESTIGATION 1: INTERFERENCE EFFECTS WITH MICROWAVES

Activity 1-1: Polarization Of Microwaves Reflected From A

Wire Grid

For this activity, you will need the following:



• Goniometer

• Wire Grid

IMPORTANT: It is imperative that you NOT peg the meter as

doing so can damage it! If you find the meter pegged,

immediately turn down the sensitivity and/or move the receiver

away from the microwave generator!

Receiver

Generator

Figure 6

1. Refer to Figure 6. Place the generator on the main arm of the

goniometer such that entrance to the “horn” is between

20-30 cm from the goniometer’s hinge (at the center of the circle).

Place the receiver on the goniometer’s shorter arm so that its horn

is about the same distance from the hinge.

2. Set the angle between the two arms the goniometer at 180° (so

that the receiver is "looking" directly at the transmitter). Set both

the receiver and the transmitter at 0° (so that the electric field of

the emitted and detected waves is oriented vertically). Turn on

the generator by plugging the AC adapter wire into the generator

and then plug the adapter into an AC outlet. Adjust the

sensitivity on the receiver to obtain a signal that is 75% of full

scale on the meter. Again, do not let the meter peg! Leave

the sensitivity at this setting for the remainder of this

Investigation.



3. Place the wire grid so that the center pin of the goniometer

hinge fits into the recess in the bottom of the grid frame (see

Figure 7). Align the grid so that it is at 40° on the goniometer

scale.

Question 1-1: What is the angle (labeled “Angle of Incidence” in

Figure 7) between the incident microwave beam and the normal to

the plane formed by the wire grid array?

Angle of Incidence

Figure 7

Prediction 1-1: At what angle between the goniometer arms do we

expect to find a maximum detected signal? Explain.

Goniometer Angle

Note: You may find it easier to slide the receiver if you put a bit of

paper under the receiver to reduce sliding friction.

4. Slide the receiver (still attached to the short goniometer arm) until

you find the angle where the detected signal is at a maximum.

[Note: You may find it easier to slide the receiver if you put a bit

of paper under the receiver to reduce sliding friction.] Record the

angle between the two goniometer arms and the detected signal

strength.

Goniometer Angle

Angle of

Incidence Angle of

Reflectance

Wire Grid

Transmitter

Receiver

40°



Question 1-2: Based on your observations, does the wire grid array

behave like a mirror? Explain.

Activity 1-2: Two Slit Interference Pattern

To further observe interference with microwaves, you will need the

following:



• Goniometer

• Double slit hood

• Meter stick or tape measure, plastic ruler

CAUTION: DO NOT ALLOW THE RECEIVER’S METER

TO PEG AT ANY TIME!

Receiver

Generator

Figure 8

1. Slide the double slit hood over the generator’s horn, creating two

coherent microwave sources as shown schematically in Figure 9.



x ≈ d sin θ

Microwave

generator

θ

θ

Horn

Double slit hood Receiver

d

Figure 9

2. Place the generator on the main arm of the goniometer such that

the hood lies directly over the goniometer’s hinge. Place the

receiver on the goniometer’s shorter arm so that the horns are

about 25 cm apart.

The signal amplitude that the receiver will detect depends on the

phase of the microwaves when they reach the receiver probe. To a

good approximation, if x ≈ d sin θ is equal to an integral number of

wavelengths nλ, then the microwaves from the two slits will

interfere constructively and you will see a maximum register on

the meter. Likewise, if d sin θ is equal to a half-integral number of

wavelengths (n - ½) λ, the meter will register a minimum.

constructive interference: sinn dλ θ=

destructive interference: ( )1 2 sinn dλ θ− =

3. Record the distance between centers of slits (double slit hood)

d: __________

4. Adjust the horn around θ = 0º to obtain a maxima signal. Then move the horn receiver to greater angles and note further

maxima and minima. You will need to increase the sensitivity to find the minima accurately. However, reduce the sensitivity as

you move away from the minima so that you do not peg the meter! You should be able to locate a minimum and a maximum

on either side of the central maximum (0º). [Remember, it is easier to slide the receiver if you place a sheet of paper under its

feet.]

Angles of minima: _____________ ______________

Angles of maxima: _____________ ______________



5. Use your data for the minima to find the wavelength. Show your calculations below.

λ: __________________

6. Use your data for the first non-central maxima to find the

wavelength. Show your calculations below.

λ: _________________

Question 1-3: How do your values compare with the given microwave wavelength (28.5 mm)? Discuss any uncertainties.

7. Turn off the receiver and set it aside.

Activity 1-3: Standing Waves

When waves moving in a given medium have the same frequency, it is possible for the waves to interfere and form a stationary

pattern called a standing wave. Standing waves, though they are not found in all waves, do occur in a variety of situations, most

familiarly perhaps in waves on a string, like in a guitar or violin. The incident and reflected waves combine according to the

superposition principle and can produce a standing wave.

We have seen how a grid of wires acts like a mirror for

microwaves. A metal plate can be thought of as the limit as the spacing between the wires vanishes. Microwaves reflected from a

metal plate have the same frequency and wavelength as the incident microwaves, but they travel away from the plate and their

phase is such that they add with the incident wave so as to cancel at the plate. At certain distances away from the plate (even

number of quarter-wavelengths, such as 2λ/4, 4λ/4, 6λ/4, …), the electric fields of the two waves will again destructively interfere

and produce a minimum signal in the detector probe, while at other



locations (odd number of quarter-wavelengths, such as λ/4, 3λ/4, 5λ/4, …) they will constructively interfere and produce a

maximum signal.

Consider the configuration shown in Figure 10 (below). The

incoming field from the generator will be reflected from the metal plate and subsequently interfere with the incident wave. We can

use the detector to find the positions of the maxima and minima and determine the wavelength of the electromagnetic field.

Figure 10

For this activity, you will need the following materials:


• Microwave probe


• Metal reflector plate

• Goniometer

• Component holders

• Meter stick or tape measure, plastic ruler

1. Locate the microwave detector probe, a rectangular piece of

circuit board (and attached cord) to which a detector diode (an electrical device that conducts current in only one direction) is

soldered. Notice the solder line that extends beyond the ends of the diode and acts as an antenna.

The antenna is designed to be equal to the length of two wavelengths, i.e. 5.7 cm. When the electric field of the microwave strikes it, an ac

voltage at a frequency 10.5 GHz is induced across the diode.

The amplitude of the DC signal from the detector diode is generally

quite weak, so it must be amplified.



2. Plug the diode cable into the jack on the side of the microwave receiver. Make sure that the receiver is pointed away so that the

horn does not “see” any of the signal.

3. Position the diode at about 50 cm from the front of the

microwave generator’s horn and oriented vertically, as shown in Figure 10. Make sure the orientation of the generator is 0º.

4. Adjust the distance of the probe from the generator until the meter registers a voltage about 3/4 of full scale. Keep the probe

at least 15 cm away from the generator to keep the diode from burning out. [The stand holding the detector probe is

easier to slide if you put a piece of paper under its feet. Also, remember to keep your hand out of the way since any conductor

in the vicinity, e.g. a piece of metal, even your hand, will reflect waves and may give you spurious results.

Prediction 1-2: If we place a reflector behind the detector probe, the microwave should be reflected back towards the generator. What do

you think will happen to the original wave and the reflected wave? What are the conditions to produce a maximum constructive standing

wave? What are the conditions to produce a minimum?

5. Place a reflector (solid flat piece of metal) behind the detector probe, as shown in Figure 10. This will produce a standing wave

between the generator and the reflector.

6. Position the probe near the plate (at least 50 cm from the

generator) and slide it along the leg of the goniometer. Notice that there are positions of maxima and minima signal strength.

Slide the detector probe along the goniometer, no more than a cm or two, until you determine a maximum signal. Then slide the

reflector, again no more than a centimeter or two, until you obtain another signal strength maximum. Continue making

slight adjustments to the detector probe and reflector until the meter reading is as high as possible, but not pegging on the 10 V

scale. If this occurs, move the generator back further away.

7. Now find a node (minimum) of the standing wave pattern by

slightly moving the probe until the meter reading is a minimum. We want to determine the wavelength of the standing wave, so

only relative distances between maxima and/or minima are relevant. In this case, it is easiest to use the goniometer scale and



measure the distance using the probe base and goniometer scale. Record the position of the probe below:

Initial probe position at minimum: ____________________

8. While watching the meter, slide the probe along the goniometer

until the probe has passed through at least ten antinodes (maxima) and returned to a node. Be sure to count the number

of antinodes that were traversed. Record the number of antinodes traversed and the new probe position.

Antinodes traversed: ______

Final probe position at minimum: ________

Question 1-4: What are the analogies with the nodes and antinodes found here and for the standing waves found from an oscillating

string fixed at both ends (guitar)? Sketch a picture.

Question 1-5: What is the distance in terms of the wavelength between adjacent antinodes (maxima)?

Question 1-6: What is the wavelength you deduce from your data?

Show your work.

λ : ________________________



Question 1-7: Using this experimental wavelength value, determine the frequency of the wave. Show your work and discuss the

agreement with what you expect.

f : ____________

9. Unplug the generator’s power supply from the receptacle and turn off the amplifier before proceeding.

INVESTIGATION 2: INTERFERENCE EFFECTS WITH VISIBLE LASER

LIGHT

Please read Appendix D: Lasers before you come to lab.

OVERVIEW

In an earlier experiment you studied various interference phenomena with electromagnetic waves whose wavelength λ was

approximately 3 cm (microwaves). In this experiment you will study similar phenomena with electromagnetic waves in the visible

part of the spectrum. For brevity, we will simply use the common term “light”. Light waves have a much shorter wavelength

(λ ≈ 4 - 7 × 10-5

cm) than do microwaves.

All the phenomena that you will observe can be described quite

accurately with a simple theoretical model dating back to Christian Huygens (1629 - 1695). This model applies to wave phenomena in

general and does not make any reference to the electromagnetic nature of light.

Huygens’ Principle states that every point of a wave front can be thought of as the origin of spherical waves. This seems to be

contradicted by experience: How can a laser emit a pencil beam? Would the light not spread out immediately?



Figure 11

We investigate this question with a gedanken (thought)

experiment: Let a plane wave be incident on a screen that has a hole cut into it. Imagine that at the time t = 0 the wave front,

coming from below, has just reached the hole as shown in Figure 11a. An instant later, at t = t1, the spherical wavelets that

were, according to Huygens’ Principal, created at every point of this wave front have begun to spread, as in Figure 11b. Later yet at

t = t2, shown in Figure 11c, we find that the wavelets have spread considerably; but, in a central region as wide as the slit, they have

formed a new wave front propagating in the same direction as the old one. This reasoning can be repeated point by point in space

and time as needed.

As a result, one will see by and large what one would have

expected: a ray of light of the width of the slit, propagating in the original direction. Only on very close observation will one see that

a small amount of light has leaked around the corner to regions where according to a ray model it should not be. The larger the

hole, the smaller will be the fraction of the light that leaks around the corner, a process called diffraction. “Largeness” is a relative

term - large with respect to what? The only measure of length that is appropriate for this problem is the wavelength of the incident

light: When the size of the hole is large compared to the wavelength only a small part of all the light will find its way

around the edge, most of it will be in the central beam. Only if the size of the hole is small compared to the wavelength will one find

that the light spreads out spherically, as shown in Figure 12 (below). Clearly this is only a hand waving argument. A



mathematically rigorous explanation of diffraction, based on Huygens’ Principle, is credited to J. A. Fresnel, (1788-1827).

Figure 12

For this investigation, you will need the following:

• Laser

• Lab jack

• Small Support stand

• Clamp for stand

• Slide with single slits of varying widths

• Slide with multiple slits

• Diffraction grating slide

• White paper

• Board placed on table to observe patterns

• 3-m tape

THE SINGLE SLIT

Diffraction patterns can be observed with a single slit. Figure 13

(below) shows three representative wavelets emerging from a single slit. The angle has been chosen so that the path difference

between the wavelets a) and c) is equal to the wavelength λ. For a single slit such a condition will result in darkness: For each wavelet emerging from the lower half of the slit there will be one

from the upper half that is out of phase by half a wavelength that will extinguish it.



Figure 13

In the case of the single slit the condition for darkness is thus

dark

n

n Dy

w

λ= (2)

where n is any non-zero integer (n = ±1, ±2, ±3…).

A more detailed derivation gives the intensity, I, as a function of

angle, θ:

2

0 2

sin sin

( )

sin

w

I = Iw

πθ

λθ

πθ

λ

. (3)

I(θ) is shown in Figure 14a. Dark bands appear when the intensity

drops to zero. It is easy to see that yndark

correspond to such

minima (since sin y Dθ = ). Bright bands appear at the maxima

of the intensity distribution. By inspection, we can see that the intensity goes to I0 as θ goes to zero (use the small angle

approximation: sin x x≈ for very small x ). The rest of these

maxima are best found through numerical methods.



-0.1 -0.05 0 0.05 0.1

Minima corresponding

to y2dark

Minima corresponding

to y1dark

Figure 14a Figure 14b

Activity 2-1: Single Slit Interference

CAUTION: THE LASER BEAM IS VERY INTENSE. DO

NOT LOOK DIRECTLY INTO IT! DO NOT LOOK AT A

REFLECTION OF THE BEAM FROM A METALLIC

SURFACE.

1. Aim the laser at white paper placed on the board at least 1 m away, and turn it on. Note the brightness of the spot on the

screen. The parallel beam of light is only a few millimeters in diameter.

2. Clamp the slide holder containing the single slit slide in the support stand so that the slits are horizontal.

3. Use the lab jack to adjust the height of the laser so that the beam passes through the second to narrowest single slit

(w = 0.04 mm).

4. Observe the diffraction pattern on the screen. You should see

something like shown in Figure 14b. [The black “bars” in the figure represent the bright bands while the gaps represent the

dark bands.]

5. Record your observations. Approximately how wide are the

bright bands?

2y1dark 2y2

dark



Question 2-1: Discuss your observations in terms of Figure 14.

6. Measure the distance from the slit slide to the screen.

D:

7. Measure the diffraction pattern for this slit. To do this, measure from the dark band on one side of the center to the dark band on

the other side of the center light band (see Figure 14). This will give you 2yn

dark. Measure for three different bands and record in

Table 2-1.

Table 2-1

8. Use the values found for 2yndark to determine yn

dark and record

them in Table 2-1.

Question 2-2: Use the bands and the measurements you found to

find the wavelength of the laser using Equation (2). Record the values in Table 2-1. Find the average and record below.

λ

9. Move the laser so that it is incident on the wider slits of 0.08 mm, and then on 0.16 mm. Look at the intensity pattern for

each.

w n 2yndark

yndark

λ

0.04 mm

1

2

3



Question 2-3: What can you qualitatively say about how the diffraction pattern changes as the slit width goes from smaller to

larger?

Question 2-4: Why do you think that the bands would get larger or smaller (depending on your answer to Question 2-3)? Why

would the bands get closer together or further apart (also depending on your previous answer)?

THE DOUBLE SLIT

We consider now the case of two very narrow parallel slits making the following assumptions:

1. The distance d between the slits is large compared to their widths.

2. Both slits are illuminated by a plane wave e.g. light from a distant source or from a laser.

3. The incident light has a well-defined wavelength. Such light is called monochromatic, i.e. light of one color.

4. The individual slits are narrow, no more than a few hundred wavelengths wide.

Each of the two slits is the source of wavelets. Since the slits are very narrow, each is the source of just one series of concentric

cylindrical wavelets, as shown in Figure 15. Both slits are illuminated by the same plane wave, the wavelets from one slit

must, therefore, be in phase with those from the other.

Let the black rings in Figure 15 indicate the positions of positive ½

waves (maxima) at a certain moment in time. In certain directions one sees black “rays” emanating from a point half way between the

slits. In these directions the waves from the two slits will overlap and add (constructive interference). In the directions in between



the black “rays” positive half waves (black) will coincide with negative half waves (white) and the waves from the two slits will

extinguish each other (destructive interference).

Figure 15

Figure 16 gives an example of destructive interference: At the

angle θ shown in the figure, the waves from the two slits are out of step by a half wavelength. Clearly, destructive interference will

also result in all those directions for which the waves from the two slits are out of step by an odd number of half wavelengths. In

those directions in which the distances traveled by the two waves differs by an even number of half wavelengths the interference will

be constructive.



Figure 16

On a screen intercepting the light, one will therefore see alternating light and dark bands. From the positions of these bands it is easy

to determine the wavelength of the light. From Figure 16 we find

sin = 2d

λθ (4)

where d is the distance between the slits and λ is the wavelength.

We can also note that

1tandark

y =

Dθ (5)

where y1dark

is the distance of the first dark band from the center

line and D is the distance from the slit to the screen. From this we

can obtain (for small angles, tan sinθ θ≈ ) the wavelength:

1

dark2y d

Dλ ≈ (6)

Convince yourself that this can be generalized to

, 1,2,3,...dark

n2y d

n(2n - 1)D

λ ≈ = (7)

where y1dark

, y2dark

, y3dark

, etc. are the distances to the first, second,

third, etc. dark band from the centerline.



Activity 2-2: Multiple Slit Interference

1. Replace the single slit slide with the multiple slit slide.

2. Make the laser light incident upon the double slit with the slide.

Question 2-5: Discus how this pattern is different from the single

slit pattern.

3. With a pencil or pen, mark a line slightly off to the side the

position of the central ray. Then mark each of the first four dark

spots above and below the central ray position. It may be

easiest to take the white paper off the board to measure the

positions between the dark spots above and below the critical

rays. Mark on your diagram where the distances 2y1dark

, 2y2dark

,

and 2y3dark

actually are. Measure them and write them down

below.

2y1dark

___________________________

2y2dark

___________________________

2y3dark

___________________________

4. Use these measurements calculate the wavelength. Show your

work.

λ



Question 2-7: How well does this value agree with the single slit

determination?

THE DIFFRACTION GRATING

Adding more slits of the same widths and with the same slit to slit

distances will not change the positions of the light and dark bands

but will make the light bands narrower and brighter. In diffraction

gratings this is carried to extremes: many thousands of lines are

scratched into a piece of glass or a mirror surface, giving the same

effect as many thousands of slits. The “slit-to-slit” distance d is

usually made very small so that for a given wavelength λ the

distances ynbright

from the bright bands to the center become very

large.

At the same time, the thickness of the bands of monochromatic

light become narrow lines. This enables one to measure the

ynbright

’s (and hence the wavelengths) very accurately with such

gratings. The order m specifies the order of the principal maxima,

and m = 0 for the central beam at a scattering angle θ = 0. The first

bright spot on either side of the central maxima would be m = 1

scattered at angle θ1; the next spot would be m = 2 scattered at

angle θ2 and so forth.

It is therefore quite easy to calculate the wavelength of light using

a diffraction grating. The wavelength is given by the equation

sin mm dλ θ= (8)

D is the distance from the diffraction grating to the screen, and the

first maximum (m = 1) is observed at an angle θ1 from the central

ray. Then ymbright

is the distance on the screen from the central ray

to the maxima corresponding to order m. Then

tan bright

mm

y

Dθ = (9)

By using the two previous equations we can calculate the

wavelength. Equation (9) can be used to find θm, and then that

value plugged into Equation (8).



Activity 2-3: The Diffraction Grating

In this activity, you will use the diffraction grating to calculate the

wavelength of the light.

1. Find the slide that contains the diffraction grating. Printed on

the slide should be the number of lines per millimeter (N).

Record this value and then determine the distance d between

adjacent slits 1/N.

N: d:

NOTE: The grating surface is about six millimeters in from the

face of the housing.

2. Measure the distance D between the diffraction grating and the

screen very precisely. It should be about 15-20 cm, to allow for

the band separations to be as large as possible while still being

on the screen.

D:

3. Shine the light from the laser through the diffraction grating and

ensure that you can measure the maxima. You probably will

have to look at the first order, because the others will be off the

screen. It may help to turn the grating and screen horizontal to

make these measurements.

4. Measure and record the distance from the center of one

maximum to the maxima on the other side of the central ray

corresponding to the same order:

2ymbright

:

5. Record the order your used:

m:

6. Use your data to calculate the wavelength of the laser light.

λ:



Question 2-8: Discuss how well this value for the laser

wavelength agrees with your previous values. Which method

(single slit, double slit, or grating) appears to be the best? Explain

and discuss possible sources of uncertainties for each method.

A-1

APPENDIX A

SELECTED CONSTANTS

I. Fundamental Constants Cgs Mks

Speed of light ( c ) 2.99792458* × 10

10 cm • s-1

108

m • s-1

Gravitational constant (G ) 6.67 × 10-8

dyn • cm2 • g

-2 10

-11 N • m2 • kg

-2

Permeability constant ( 0µ ) 1.26 × –– 10-6

Henry • m-1

Permittivity constant ( 0ε ) 8.85 × –– 10-12

Farad • m-1

Electron charge ( e ) 1.60219 × –– 10-19

Coulomb

4.80325 × 10-10

esu

Planck’s constant ( h ) 6.6262 × 10-27

erg • s 10-34

J • s

4.1357 × 10-15

eV • s

Planck’s constant ( π= 2hℏ ) 1.05459 × 10-27 erg • s 10-34 J • s 6.5822 × 10-16 eV • s

Avogadro’s number ( AN ) 6.022 × 1023 mol-1 1023 mol-1

Boltzmann’s constant ( Bk ) 1.3807 × 10-16 erg • K-1 10-23 J • K-1

8.617 × 10-5 eV • K-1

Gas constant ( R ) 8.314 × 107 erg • K-1 mol-1 1 J • K-1 • mol-1

Bohr radius ( 0a ) 0.529177 × 10-8 cm 10-10 m

Rydberg ( R ) 1.09737 × 105 cm-1 107 m-1

13.6058 × eV/ hc

II. Other Physical Constants Cgs Mks

Acceleration of gravity ( g ) 9.80665 × 102 cm • s-2 m • s-2

local 9.809 × 102 cm • s-2 m • s-2

at equatorial sea level 9.78 × 102 cm • s-2 m • s-2

at polar sea level 9.83 × 102 cm • s-2 m • s-2

Earth’s radius (earth

r ) 6.38 × 108 cm 106 m

Earth’s mass (earth

m ) 5.98 × 1027 g 1024 kg

Electron rest mass (e

m ) 9.1095 × 10-28 g 10-31 kg

Proton rest mass (p

m ) 1.6726 × 10-24 g 10-27 kg

Neutron rest mass (n

m ) 1.6748 × 10-24 g 10-27 kg

Speed of sound in dry air at STP 3.31 × 104 cm • s-1 102 m • s-1

* This number is exact as the meter is now defined in terms of c and the second.

A-2

II. Other Physical Constants Cgs Mks (continued)

Heat capacity of water 4.19 × 107 erg • g-1 • K-1 103 J • kg-1 • K-1

Heat of fusion of water (at 100°C) 3.34 × 109 erg • g-1 105 J • kg-1

Heat of vaporization of water (at 0°C) 2.27 × 1010 erg • g-1 106 J • kg-1

Index of refraction ( n ) of

water (589.2 nm) 1.33

crown glass (589.2 nm) 1.52

air (590.0 nm) 1.0002765

III. Conversion Factors

1 Electron volt (eV) 1.60219 × 10-19 J

1 angstrom (Å) 0.1 nm

1 Pascal (Pa) 1 N • m-2

1 Torr 1 mm Hg = 133 Pa

1 atmosphere (atm) 760 Torr = 101.3 kPa

1 erg 10-7 J

1 calorie 4.18 J

1 Tesla 1 Weber • m-2 = 104 Gauss

IV. Standard Resistor Color Code

first three bands fourth band tolerance black 0 green 5 unmarked 20%

brown 1 blue 6 silver 10% red 2 violet 7 gold 5%

orange 3 gray 8 yellow 4 white 9

V. Temperatures of Substances (at a pressure of one atmosphere)

Boiling point of water 100°C = 373.15 K

Melting point of ice 0°C ≡ 273.15 K

Dry ice + methanol -78.5°C = 194.7 K

Boiling point liquid nitrogen -195.8°C = 77.4 K

Boiling point liquid helium -269.0°C = 4.2 K

VI. Color as Function Of Wavelength

400 - 450 nm Violet

450 - 500 nm Blue 500 - 575 nm Green

575 - 595 nm Yellow 595 - 620 nm Orange

620 - 700 nm Red

B-1

APPENDIX B

GRAPHICAL ANALYSIS

Introduction

There are two basic ways to present data: the data table and the graph. In this appendix,

we will try to acquaint you with some of the points to be considered in preparing a proper

graph. The discussion assumes hand drawn paper graphs, but the ideas are, of course,

applicable when using graphing software.

Selecting the Graph Paper

For a first graph, made while the data are being taken, you might find it convenient to just

use the square-ruled paper of your lab notebook. If you are like most people, you will be

able to divide a small length into five equal parts by eye with sufficient accuracy. A

typical notebook has squares ¼ × ¼ inch. Allowing some margins for labeling that

leaves you an area of about 30 × 40 squares. Counting on your ability to interpolate to

about 1/5 of the width of an individual square this will permit you to plot a graph

containing 150 × 200 units. Choose your scale so that the graph fills the page as much as

possible without, however, going to strange units. Thus, if you want to plot 100 seconds

along the 30 squares of the x -axis, you will be better off if you use units of 5 sec per

division instead of 3.333 sec/div, even though the latter choice would have filled the

available space. Do not use 8 div/inch because that forces you to use fractions instead of

the more convenient decimal scale.

For a formal lab report as well as for a better looking journal, you might want to use

regular graph paper ruled in either millimeters or 1/10 of an inch. Again, avoid the kind

of graph paper that is ruled in 1/2, 1/4, and 1/8 inches.

Logarithmic Paper

Sometimes it becomes necessary or desirable to plot the logarithm of a quantity instead

of the quantity itself:

Necessary: In the case that a parameter varies over several orders of magnitude, the

drastic compression of the scale by the logarithm ( log10 1= , log100 2= , log1000 3= ,

etc.) makes it possible to plot the data over a very wide range of the argument and/or the

function.

Desirable: A logarithmic plot brings out the functional relation between the two

variables plotted along the x- and y-axes. Consider, for example, the function

ay Cx= (1)

For a = 1 it is represented by a straight line, for any other value of a by some curve or

other. If we take the logarithm of both sides of this equation we get

log log logy C a x= + . (2)

This equation is represented by a straight line if we plot log x along the abscissa (the

x-axis) and log y along the ordinate (the y-axis) of linear graph paper. Doubly

B-2

logarithmic graph paper is ruled so that we do not need to take the logarithm, we simply

plot x along the (logarithmically ruled) x-axis and y along the (logarithmically ruled)

y-axis and get a straight line in the resulting log-log plot. The slope of that straight line is

given by the exponent a. Note that this is true only for a simple power law of the form

given by Equation 1. Even an additive constant will make the log-log plot non-linear.

If one wishes to plot an exponential function of the form

axy Ce= , (3)

it is expedient to use semi-log paper, i.e. graph paper on which the x-axis is ruled

linearly and the y-axis logarithmically. If one plots Equation 3 on such paper, a straight

line with the slope a will result. Again any additive term, even a simple constant will

make the resulting plot non-linear.

To find the slope from a logarithmic graph, read off two points from your straight line

and solve the relevant equation, solving for the slope. For example, working from

Equation 2,

1 1

2 2

log log log

log log log

y C a x

y C a x

= +

= +

, (4)

so

2 1 2 1log log (log log )y y a x x− = − , (5)

or

( )

( )

2 1

2 1

log

log

y ya

x x= , (6)

where x1, x2, y1, and y2 are read directly off the graph.

Just in case you do not remember what logarithms are all about, we list some useful

formulas and definitions:

ln natural log loge

x x x≡ = xxx 10loglogcommonlog =≡

( ) xe x=ln ( ) xx

=10log

ln 2.3026logx x= baab lnln)ln( +=

bab

alnlnln −=

( ) axa x lnln =

Plotting the Graph

Select scales for the x - and y -axes so that the graph fills the available area as much as is

practical. In a linear plot, you might consider suppressing zero. For example, if y varies

only from 100 to 125 you might want to start the y -axis at y = 100 and spread the

interval from 100 to 125 over the entire length of the axis. Plot each individual point first

with a pencil and, after a final check of all points, mark them permanently. A dot with a

concentric circle around it will both mark a point precisely and draw attention to it. If

you plot more than one curve on a graph, select different symbols for the data points of

B-3

each curve. Indicate the size of the probable uncertainty by error bars; if appropriate

show error bars in both the x - and y -directions.

One usually plots not only the measured data points on a graph but connects them also

with some kind of curve. This curve can either be a graphic representation of a theory

that (one hopes) describes the experiment or it can simply be a smooth line drawn by eye

with a French curve that more or less follows the data points. Whatever it is should be

clearly stated in the caption underneath the figure, e.g. “The curve shows the calculated

values according to … [the formula],” or “The curve was drawn to guide the eye.”

Often the slope of a curve, i.e. the tangent of the angle with the x -axis, conveys useful

information. This does not mean that you can learn anything by taking a protractor to

your curve, measuring its angle with the x -axis and then taking the tangent of that angle.

Let us assume that you have plotted a distance x (measured in meters) against the time t

(measured in seconds) it takes to travel that distance. In that case, the slope x t∆ ∆ of the

resulting curve will be a measure of the velocity. This slope, however, will depend on

the scales that you have used along the two axes. You will get the correct value of the

velocity in m/s only if you divide the value of x∆ (in meters) that you have read off the

y -scale by the value of t∆ (in seconds) read off the x -scale.

Labeling the Graph

Every graph should have a title that tells what is shown. You should also label both axes

and give the units used. The customary way is to give the name of the variable followed

by the dimension in square brackets, e.g.: time [sec]. A graph without proper labeling of

the axes gives no useful information.

An Example

Figures 1 and 2 (following) show an example of the same set of data plotted on a linear

and a log-log plot. The time it took a free falling steel sphere of 1 cm diameter to fall a

distance S was measured at 20 cm intervals. The experimental uncertainties were

smaller than the size of the data points. The curve represents the equation gSt /2=

for g = 9.8 m/s2.

B-4

Figure 1. Time as a function of distance in free fall.

Figure 2. Same as Figure 1, but plotted on log-log plot.

C-1

APPENDIX C

ACCURACY OF MEASUREMENTS AND

TREATMENT OF EXPERIMENTAL

UNCERTAINTY

“A measurement whose accuracy is unknown has no use whatever. It is there-

fore necessary to know how to estimate the reliability of experimental data and

how to convey this information to others.”

—E. Bright Wilson, Jr., An Introduction to Scientific Research

Our mental picture of a physical quantity is that there exists some unchanging, underlying

value. It is through measurements we try to find this value. Experience has shown that

the results of measurements deviate from these “true” values. The purpose of this Ap-

pendix is to address how to use measurements to best estimate the “true” values and how

to estimate how close the measured value is likely to be to the “true” value. Our under-

standing of experimental uncertainty (sometimes called errors) is based on the mathemat-

ical theory of probability and statistics, so the Appendix also includes some ideas from

this subject. This Appendix also discusses the notation that scientists and engineers use

to express the results of such measurements.

Accuracy and Precision

In common usage, “accuracy” and “precision” are synonyms. To scientists and engi-

neers, however, they refer to two distinct (yet closely related) concepts. When we say

that a measurement is “accurate”, we mean that it is very near to the “true” value. When

we say that a measurement is “precise”, we mean that it is very reproducible. [Of course,

we want to make accurate AND precise measurements.] Associated with each of these

concepts is a type of uncertainty.

Systematic uncertainties are due to problems with the technique or measuring instrument.

For example, as many of the rulers found in labs have worn ends, length measurements

could be wrong. One can make very precise (reproducible) measurements that are quite

inaccurate (far from the true value).

Random uncertainties are caused by fluctuations in the very quantities that we are meas-

uring. You could have a well calibrated pressure gauge, but if the pressure is fluctuating,

your reading of the gauge, while perhaps accurate, would be imprecise (not very repro-

ducible).

Through careful design and attention to detail, we can usually eliminate (or correct for)

systematic uncertainties. Using the worn ruler example above, we could either replace

the ruler or we could carefully determine the “zero offset” and simply add it to our rec-

orded measurements.

Random uncertainties, on the other hand, are less easily eliminated or corrected. We

usually have to rely upon the mathematical tools of probability and statistics to help us

determine the “true” value that we seek. Using the fluctuating gauge example above, we

C-2

could make a series of independent measurements of the pressure and take their average

as our best estimate of the true value.

Probability

Scientists base their treatment of random uncertainties on the theory of probability. We

do not have space or time for a lengthy survey of this fundamental subject, but can only

touch on some highlights. Probability concerns random events (such as the measure-

ments described above). To some events we can assign a theoretical, or a priori, proba-

bility. For instance, the probability of a “perfect” coin landing heads or tails is 12

for each

of the two possible outcomes; the a priori probability of a “perfect” die* falling with a

particular one of its six sides uppermost is 16

.

The previous examples illustrate four basic principles about probability:

1. The possible outcomes have to be mutually exclusive. If a coin lands heads, it

does not land tails, and vice versa.

2. The list of outcomes has to exhaust all possibilities. In the example of the coin

we implicitly assumed that the coin neither landed on its edge, nor could it be

evaporated by a lightning bolt while in the air, or any other improbable, but not

impossible, potential outcome. (And ditto for the die.)

3. Probabilities are always numbers between zero and one, inclusive. A probability

of one means the outcome always happens, while a probability of zero means the

outcome never happens.

4. When all possible outcomes are included, the sum of the probabilities of each ex-

clusive outcome is one. That is, the probability that something happens is one. So

if we flip a coin, the probability that it lands heads or tails is 1 12 2

1+ = . If we toss

a die, the probability that it lands with 1, 2, 3, 4, 5, or 6 spots showing is 1 1 1 1 1 16 6 6 6 6 6

1+ + + + + = .

The mapping of a probability to each possible outcome is called a probability distribu-

tion. Just as our mental picture of there being a “true” value that we can only estimate,

we also envision a “true” probability distribution that we can only estimate through ob-

servation. Using the coin flip example to illustrate, if we flip the coin four times, we

should not be too surprised to get heads only once. Our estimate of the probability distri-

bution would then be 14

for heads and 34

for tails. We do expect that our estimate would

improve as the number of flips† gets “large”. In fact, it is only in the limit of an infinite

number of flips that we can expect to approach the theoretical, “true” probability distribu-

tion.

* …one of a pair of dice.

† Each flip is, in the language of statistics, called a trial. A scientist or engineer would probably

say that it is a measurement or observation.

C-3

A defining property of a probability distribution is that its sum (integral) over a range of

possible measured values tells us the probability of a measurement yielding a value with-

in the range.

The most common probability distribution encountered in the lab is the Gaussian distri-

bution. The Gaussian distribution is also known as the normal distribution. You may

have heard it called the bell curve (it is shaped somewhat like a fancy bell) when applied

to grade distributions.

The mathematical form of the Gaussian distribution is:

2 2

2

21

2( ) d

GP d e− σ

πσ

= (1)

The Gaussian distribution is ubiquitous because it is the

end result you get if you have a number of processes,

each with their own probability distribution, that “mix

together” to yield a final result. We will come back to

probability distributions after we've discussed some sta-

tistics.

Statistics

Measurements of physical quantities are expressed in numbers. The numbers we record

are called data, and numbers we compute from our data are called statistics. A statistic is

by definition a number we can compute from a set of data.

Perhaps the single most important statistic is the mean or average. Often we will use a

“bar” over a variable (e.g., x ) or “angle brackets” (e.g., x ) to indicate that it is an aver-

age. So, if we have N measurements i

x (i.e., 1x , 2x , ..., N

x ), the average is given by:

1 2

1

1( ... ) /

N

N i

i

x x x x x N xN

=

≡ = + + + = ∑ (2)

In the lab, the average of a set of measurements is usually our best estimate of the “true”

value*:

x x≈ (3)

In general, a given measurement will differ from the “true” value by some amount. That

amount is called a deviation. Denoting a deviation by d , we then obtain:

i i i

d x x x x= − ≈ − (4)

Clearly, the average deviation ( d ) is zero (to see this, take the average of both sides). It

is not a particularly useful statistic.

A much more useful statistic is the standard deviation, defined to be the “root-mean-

square” (or RMS) deviation:

* For these discussions, we will denote the “true” value as a variable without adornment (e.g., x).

0

C-4

2 2

1

1( )

N

x i

i

d x xN

σ

=

= = −∑ (5)

The standard deviation is useful because it gives us a measure of the spread or statistical

uncertainty in the measurements.

You may have noticed a slight problem with the expression for the standard deviation:

We don't know the “true” value x , we have only an estimate, x , from our measurements.

It turns out that using x to instead of x in Equation (5) systematically underestimates the

standard deviation. It can be shown that our best estimate of the “true” standard devia-

tion is given by the sample standard deviation:

2

1

1( )

1

N

x i

i

s x xN

=

= −

− ∑ (6)

To illustrate some of these points, consider the following: Suppose we want to know the

average height and associated standard deviation of the entering class of students. We

could measure every entering student and simply calculate the average. We would then

simply calculate x and x

σ directly. Tracking down all of the entering students, however,

would be very tedious. We could, instead, measure a representative* sample and calcu-

late x and x

s as estimates of x and x

σ .

Modern spreadsheets (such as MS Excel) as well as some calculators (such as HP and

TI) also have built-in statistical functions. For example, AVERAGE (Excel) and x (cal-

culator) calculate the average of a range of cells; whereas STDEV (Excel) and xs (calcu-

lator) calculate the sample standard deviations.

Standard Error

We now return to probability distributions. Consider Equation (1), the expression for a

Gaussian distribution. You should now have some idea as to why we wrote it in terms of

d and σ. Most of the time we find that our measurements (xi) deviate from the “true” val-

ue (x) and that these deviations (di) follow a Gaussian distribution with a standard devia-

tion of σ. So, what is the significance of σ? Remember that the integral of a probability

distribution over some range gives the probability of getting a result within that range. A

straightforward calculation shows that the integral of PG [see Equation (1)] from -σ to +σ

is about 23

. This means that there is probability of 23

for any single† measurement being

within ±σ of the “true” value. It is in this sense that we introduce the concept of standard

error.

Whenever we report a result, we also want to specify a standard error in such a way as to

indicate that we think that there is roughly a 23

probability that the “true” value is within

* You have to be careful when choosing your sample. Measuring the students who have basket-

ball scholarships would clearly bias your results. In the lab we must also take pains to ensure that

our samples are unbiased. † We'll come back to the issue of the standard error in the mean.

C-5

the range of values between our result minus the standard error to our result plus the

standard error. In other words, if x is our best estimate of the “true” value x and xσ is

our best estimate of the standard error in x , then there is a 23

probability that:

x xx x xσ σ− ≤ ≤ +

When we report results, we use the following notation:

xx σ±

Thus, for example, the electron mass is given in the 2006 Particle Physics Booklet as

me = (9.1093826 ± 0.0000016) × 10-31

kg.

By this we mean that the electron mass lies between 9.1093810×10-31

kg and

9.1093842×10-31

kg, with a probability of roughly 23

.

Significant Figures

In informal usage the least significant digit implies something about the precision of the

measurement. For example, if we measure a rod to be 101.3 mm long but consider the

result accurate to only ±0.5 mm, we round off and say, “The length is 101 mm.” That is,

we believe the length lies between 100 mm and 102 mm, and is closest to 101 mm. The

implication, if no error is stated explicitly, is that the uncertainty is ½ of one digit, in the

place following the last significant digit.

Zeros to the left of the first non-zero digit do not count in the tally of significant figures.

If we say U = 0.001325 Volts, the zero to the left of the decimal point, and the two zeros

between the decimal point and the digits 1325 merely locate the decimal point; they do

not indicate precision. [The zero to the left of the decimal point is included because dec-

imal points are small and hard to see. It is just a visual clue—and it is a good idea to pro-

vide this clue when you write down numerical results in a laboratory!] The voltage U has

thus been stated to four (4), not seven (7), significant figures. When we write it this way,

we say we know its value to about ½ part in 1,000 (strictly, ½ part in 1,325 or one part in

2,650). We could bring this out more clearly by writing either U = 1.325×10-3

V, or

U = 1.325 mV.

Propagation of Errors

More often than not, we want to use our measured quantities in further calculations. The

question that then arises is: How do the errors “propagate”? In other words: What is the

standard error in a particular calculated quantity given the standard errors in the input

values?

Before we answer this question, we want to introduce a new term: The relative error of a

quantity Q is simply its standard error, Q

σ , divided by the absolute value of Q. For ex-

ample, if a length is known to 49±4 cm, we say it has a relative error of 4/49 = 0.082. It

C-6

is often useful to express such fractions in percent*. In this case we would say that we

had a relative error of 8.2%.

We will simply give the formulae for propagating errors† as the derivations are a bit be-

yond the scope of this exposition.

1. If the functional form of the derived quantity ( f ) is simply the product of a con-

stant (C ) times a quantity with known standard error ( x and x

σ ), then the stand-

ard error in the derived quantity is the product of the absolute value of the

constant and the standard error in the quantity:

( ) f xf x Cx Cσ σ= → =

2. If the functional form of the derived quantity ( f ) is simply the sum or difference

of two quantities with known standard error ( x and x

σ and y and y

σ ), then the

standard error in the derived quantity is the square root of sum of the squares of the errors:

2 2

( , ) or ( , ) f x yf x y x y f x y x y σ σ σ= + = − → = +

3. If the functional form of the derived quantity ( f ) is simply the product or ratio of

two quantities with known standard error ( x and x

σ and y and y

σ ), then the rel-

ative standard error in the derived quantity is the square root of sum of the squares of the relative errors:

2 2

( , ) or ( , ) / | | ( / ) ( / )f x yf x y x y f x y x y f x yσ σ σ= × = → = +

4. If the functional form of the derived quantity ( f ) is a quantity with known stand-

ard error ( x and x

σ ) raised to some constant power (a ), then the relative stand-

ard error in the derived quantity is the product of the absolute value of the constant and the relative standard error in the quantity:

( ) / | | / | |a

f xf x x f a xσ σ= → =

5. If the functional form of the derived quantity ( f ) is the log of a quantity with

known standard error ( x and x

σ ), then the standard error in the derived quantity

is the relative standard error in the quantity:

( ) ln( ) /f xf x x xσ σ= → =

* From per centum, Latin for “by the hundred”.

† Important Note: These expressions assume that the deviations are small enough for us to ignore

“higher order” terms and that there are no correlations between the deviations of any of the quan-

tities x , y , etc.

C-7

6. If the functional form of the derived quantity ( f ) is the exponential of a quantity

with known standard error ( x and x

σ ), then the relative standard error in the de-

rived quantity is the standard error in the quantity:

( ) /x

f xf x e fσ σ= → =

7. A commonly occurring form is one the product of a constant and two quantities

with known standard errors, each raised to some constant power. While one can successively apply the above formulae (see the example below), it is certainly easier to just use:

22

( ) /ya b x

f

baf x Cx y f

x y

σσσ

= → = +

And, finally, we give the general form (you are not expected to know or use this equa-

tion; it is only given for “completeness”):

22

2 2 2( , ,...) ...f x y

f ff x y

x yσ σ σ

∂ ∂ → = + +

∂ ∂ (7)

Standard Error in the Mean

Suppose that we make two independent measurements of some quantity: x1 and x2. Our

best estimate of x, the “true value”, is given by the mean, 11 22

( )x x x= + , and our best es-

timate of the standard error in x1 and in x2 is given by the sample standard deviation,

( ) ( ) ( )1 2

2 211 22 1x x xs x x x xσ σ

−

= = = − + −

. Note that sx is not our best estimate of xσ ,

the standard error in x . We must use the propagation of errors formulas to get xσ .

Now, x is not exactly in one of the simple forms where we have a propagation of errors

formula. However, we can see that it is of the form of a constant, ( )12

, times something

else, 1 2( )x x+ , and so:

1 2

12x x xσ σ

+=

The “something else” is a simple sum of two quantities with known standard errors (x

s )

and we do have a formula for that:

1 2 1

2 2 2 2

2 2x x x x x x xs s sσ σ σ+

= + = + =

So we get the desired result for two measurements:

1

2x xsσ =

C-8

By taking a second measurement, we have reduced our standard error by a factor of 12.

You can probably see now how you would go about showing that adding third, 3x ,

changes this factor to 13. The general result (for N measurements) for the standard

error in the mean is:

1x xN

sσ = (8)

Example

We can measure the gravitational acceleration g near the Earth’s surface by dropping a

mass in a vertical tube from which the air has been removed. Since the distance of fall

(D), time of fall (t) and g are related by D = ½ gt2, we have g = 2D/t2. So we see that we

can determine g by simply measuring the time it takes an object to fall a known distance.

We hook up some photogates* to a timer so that we measure the time from when we re-lease a ball to when it gets to the photogate. We very carefully use a ruler to set the dis-

tance (D) that the ball is to fall to 1.800 m. We estimate that we can read our ruler to within ±1 mm. We drop the ball ten times and get the following times (ti): 0.6053,

0.6052, 0.6051, 0.6050, 0.6052, 0.6054, 0.6053, 0.6047, 0.6048, and 0.6055 seconds.

The average of these times ( )t is 0.605154 seconds. Our best estimate of g is then

2

exp 2 / 9.8305g D t= = m/s2. This is larger than the “known local” value of 9.809 m/s2

by 0.0215 m/s2 (0.2%). We do expect experimental uncertainties to cause our value to be

different, but the question is: Is our result consistent with the “known value”, within ex-perimental uncertainties? To check this we must estimate our standard error.

VERY IMPORTANT NOTE: Do NOT round off intermediate results when making calculations. Keep full “machine precision” to minimize the effects of round-off errors. Only round off final results and use your error estimate to guide you as to how many dig-

its to keep.

Our expression for g is, once again†, not precisely in one of the simple propagation of

errors forms and so we must look at it piecemeal. This time we will not work it all out

algebraically, but will instead substitute numbers as soon as we can so that we can take a look at their effects on the final standard error.

What are our experimental standard errors? We've estimated that our standard error in

the distance (D

σ ) is 1 mm (hence a relative error, D Dσ , of 0.000556 or 0.0556%).

From our time data we calculate the sample standard deviation (t

s ) to be

0.000259 seconds. Recall that this is not the standard error in the mean (our best estimate of the “true” time for the ball to fall), it is the standard error in any single one of the time

measurements (i

t ). The standard error in the mean is st divided by the square root of the

* A device with a light source and detector that changes an output when something comes be-

tween the source and detector. † Refer to the discussion of the standard error in the mean.

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number of samples (10): / 10t tsσ = = 0.0000819 seconds (for a relative error, /t tσ ,

of 0.000135 or 0.0135%).

We see that the relative error in the distance measurement is quite a bit larger than the

relative error in the time measurement and so we might assume that we could ignore the

time error (essentially treating the time as a constant). However, the time enters into g

as a square and we expect that that makes a bigger contribution than otherwise. So we don’t (yet) make any such simplifying assumptions.

We see that our estimate of g (which we denote by exp

g ) is of the form of a constant (2)

times something else ( 2/D t ) and so:

2exp /

2g D tσ σ=

2/D t is of the form of a simple product of two quantities ( D and 2

t ) and so:

( ) ( )2 2

222 2

// / / /DD t t

D t D tσ σ σ= +

Now we are getting somewhere as we have D Dσ (0.000556). We need only

find 2

2/

ttσ .

2t is of the form of a quantity raised to a constant power and so:

2

2/ 2 / 0.000271

ttt tσ σ= =

Now we can see the effect of squaring t : Its contribution to the standard error is doubled.

Consider the two terms under the square root:

( ) ( )2

22 7 2 8/ 3.09 10 and / 7.33 10D t

D tσ σ− −

= × = ×

Now we can see that, even though the time enters as a square, we would have been justi-

fied in ignoring its contribution to the standard error in g. Plugging the numbers back in,

we finally get exp

0.00608g

σ = m/s2.

We see that our result is 3.5 standard deviations larger than the “known value”. While

not totally out of the question, it is still very unlikely and so we need to look for the

source of the problem. In this case we find that the ruler is one of those with a worn end.

We carefully measure the offset and find the ruler to be 5.0 mm short. Subtracting this

changes D to 1.795 m and gexp to 9.803 m/s2, well within our experimental error*.

* The implied error in our measurement of the offset (0.05 mm) is much smaller than the error in

the original D and so we can afford to ignore its contribution to the standard error in gexp.

C-10

Summary of Statistical Formulae

Sample mean: 1

1 N

i

i

x xN

=

= ∑

(best estimate of the “true” value of x , using N measurements)

Sample standard deviation: 2

1

1( )

1

N

x i

i

s x xN

=

= −

− ∑

(best estimate of error in any single measurement, i

x )

Standard error of the mean: 1

x xs

Nσ =

(best estimate of error in determining the population mean, x )

Summary of Error Propagation Formulae*

Functional form Error propagation formula

1. ( )f x Cx= ........................................................ f xCσ σ=

2. ( , )f x y x y= ± .......................................... 2 2

f x yσ σ σ= +

3. ( , ) orf x y x y x y= × ................................ 2 2

/ | | ( / ) ( / )f x yf x yσ σ σ= +

4. ( ) af x x= .................................................... / | | / | |f xf a xσ σ=

5. ( ) ln( )f x x= ................................................ /f x xσ σ=

6. ( ) xf x e= ..................................................... /f xfσ σ=

7. ( ) a bf x Cx y= ..................................................

22

/yx

f

baf

x y

σσσ

= +

and the general form:

8. ( , ,...)f x y ....................................................

22

2 2 2 ...f x y

f f

x yσ σ σ

∂ ∂ = + +

∂ ∂

* These expressions assume that the deviations are small enough for us to ignore “higher order”

terms and that there are no correlations between the deviations of any of the quantities x , y , etc.

D-1

APPENDIX D

LASERS

The wavelength λ of light is related to the frequency ν of a light wave through

ν

λc

= , (1)

where c is the velocity of light. You may recall that, according to quantum mechanics, light

consists of individual particles, photons, whose energy E is connected with the frequency ν of the

light wave through:

λ

νhc

EhE == hence, , (2)

where h = 6.63 × 10-34

[J·s] is a fundamental constant of nature, Planck’s constant.

You may also recall that atoms can exist only in states with certain, well defined, energies. If an

atom is in its state of lowest energy, the ground state, one can lift it into one of the states of higher

energy, an excited state, by bombarding it with photons whose energy νhE = is exactly equal to

the energy difference E∆ between the ground state and the excited state.

Once in an excited state, an atom will usually decay rapidly to the ground state by emitting another

photon of the same energy EE ∆= . It was Albert Einstein who realized that an atom that is already

in an excited state can be de-excited by a photon of the proper energy EE ∆= . In going back down

to the ground state it will, of course, emit another photon of that same energy.

Is it, perhaps, possible to design an amplifier for light based on this process? You see how it might

work: take a large number of atoms in an excited state and shoot in a photon of just the right energy.

This photon will de-excite one of the atoms creating another photon. Now we have two photons of

the same energy which can hit two atoms and de-excite them creating two more photons which in

turn, etc. etc. This should work all the better since the secondary photons share with the original

ones not only the energy, they also have the same phase and travel in the same direction.

Normally atoms are in their ground state while the excited states are unoccupied. To make this

scheme work one needs to create a population inversion: lots of atoms in the upper and few in the

lower state.

For decades after Einstein’s discovery physicists believed that ‘one could show’ that such an

inversion could not be accomplished. According to Einstein, those atoms that can be easily excited

will easily decay to the ground state by spontaneous emission. Those that do tend to linger in the

excited state are those that are difficult to excite in the first place.

If something is really forbidden by a law of nature it just cannot happen. However, it is often the

case that it is not really forbidden, just that people have just not been clever enough. Such is the case

here. Several ways are now known to put atoms into an excited state and keep them there until they

can be de-excited by incident photons. The good example is the well-known helium-neon gas laser

and we shall discuss it briefly.

The helium atom has an excited state that is metastable: it lives about thousand times longer than

excited atomic states normally do. According to Einstein, it should be a thousand, or so, times

harder to excite than other states, and it is harder to do so ... by photons. However, it can quite

D-2

readily be excited by bombarding the He atom with electrons, as in a gas discharge. It is thus

possible to produce large numbers of metastable He atoms in the so-called 2S1 state (see Figure 1).

By the luckiest of coincidences, the neon atom has a long-lived excited state, the 3S2 state, that has

almost exactly the same energy as the metastable He state. When an excited He atom collides with a

Ne atom in its ground state, it quite often transfers its energy to the Ne atom, returning itself to the

ground state while leaving the latter in the excited 3S2 state, as shown in Figure 1. When hit by a

photon with a wavelength of 632.8 nm the Ne atom will be de-excited to the lower, but still excited

2P4 state. It is between the 3S2 and the 2P4 states of neon that the laser action takes place.

Transfer by

collisions 2S

2P4

Excitation by

collisions with

electrons

En

erg

y

Laser light

De-excitation by

spontaneous and

induced emission

λ = 632.8 nm

Ground states

Helium Neon

3S2

Figure 1 How a HeNe laser works.

Why between these two states? There are many atoms in the Ne ground state so that it will be quite

impossible to pump enough Ne atoms into the 3S2 state to achieve a population inversion. The 2P4

state of neon, being an excited state itself, is empty.

To make a laser really work one needs a good bit of amplification, i.e. the photons must encounter a

large number of atoms in the upper state. This can be accomplished by either making the laser tube

very long or, more practically, by putting a mirror at either end, making the photons bounce back

and forth many times. This requires a great deal of precision: the secondary photons are in phase

with the primary ones, and must remain so upon reflection by the mirrors. This requires that the

latter are an integer number of wavelengths apart. In other words, a laser tube is a highly precise

interferometer.

1The labels 2S, 3S2, etc. give the experts detailed information about the quantum state.

Documents

Name: Letter: PHYS 2419 - University of Virginiapeople.virginia.edu/~ecd3m/2419/Fall2011/manual/manual.pdfPHYSICS 2419 WORKSHOP MANUAL Contents Page Introduction 1 Lab 1 Electrostatics