Physics with Calculus IInwsccmoodledemo.com/thoward/214PowerPoint.pdf · 2015-07-23 · [25.16] It takes 385 nJ of energy to push the charges apart. I’m doing work to push them

Physics with Calculus IIPhysics with Calculus II

Revised: Revised: 1/9/20141/9/2014

Where weWhere we’’ve been:ve been:

Mechanics, which includes:Mechanics, which includes:

•• MotionMotion•• NewtonNewton’’s Laws of Motions Laws of Motion•• Work & PowerWork & Power•• Rotational MotionRotational Motion•• Static EquilibriumStatic Equilibrium•• Law of GravitationLaw of Gravitation•• Fluid MechanicsFluid Mechanics•• Thermodynamics Thermodynamics

Ch 23 IntroductionCh 23 Introduction

Here we begin a study of the electric forces and the electric Here we begin a study of the electric forces and the electric field. In daily life we find the electrical force operating field. In daily life we find the electrical force operating everywhere, because it is the force that holds most things everywhere, because it is the force that holds most things together. In particular, the electric force binds atoms together. In particular, the electric force binds atoms together as in molecules or even as liquids and solids. The together as in molecules or even as liquids and solids. The force of gravity is also common, but it is only capable of force of gravity is also common, but it is only capable of holding objects together when at least one is very massive. holding objects together when at least one is very massive. On the other hand, the electrical force acts on the atomic On the other hand, the electrical force acts on the atomic level in binding electrons to the atomic nucleus and level in binding electrons to the atomic nucleus and especially in the formation of molecules, and is the force especially in the formation of molecules, and is the force most responsible for chemical reactions. most responsible for chemical reactions.

Where weWhere we’’re going:re going:

Electricity and MagnetismElectricity and Magnetism


Ch 23: Electric FieldsCh 23: Electric Fields

(23.1) Properties of electric charges (23.1) Properties of electric charges (23.2) Insulators and conductors(23.2) Insulators and conductors(23.3) Coulomb(23.3) Coulomb’’s Laws Law(23.4) The Electric Field ( E )(23.4) The Electric Field ( E )(23.5) E of continuous charge distribution(23.5) E of continuous charge distribution(23.6) Electric Field Lines(23.6) Electric Field Lines(23.7) Motion of charges in Electric Field (23.7) Motion of charges in Electric Field

(23.1) Properties of electric charges(23.1) Properties of electric charges

•• Charge is quantizedCharge is quantized•• Charge is conserved (transfer body to body)Charge is conserved (transfer body to body)•• Unlike charges attract; like charges repelUnlike charges attract; like charges repel

•• Force between chargesForce between charges2

1r

∝

(23.2) Insulators and conductors(23.2) Insulators and conductors

ConductorConductor: readily allows charges to flow: readily allows charges to flow

InsulatorInsulator: inhibits charge flow : inhibits charge flow

(23.3) Coulomb(23.3) Coulomb’’s Laws Law

((--) ) ≡≡ attractionattraction(+) (+) ≡≡ repulsionrepulsion

1 22

q qF kr

=2

928.99(10 )

NmkC

=

Applies ONLY to point chargesApplies ONLY to point charges

Not directionNot direction

1 22gravity

m mF Gr

=


181 Coulomb 1C = 6.28 (10 ) electron≡

So, one electron has the value:So, one electron has the value:

1918

1 1.60(10 )6.28(10 )

C Ceelecton electron

−= − = −

electron charge electron charge ≡≡ negativenegative

Fundamental ChargeFundamental Charge


““ee”” is the charge of the electron is the charge of the electron ANDAND the proton.the proton.

-19e = 1.60 (10 ) Coulomb

Problem [23.15] Problem [23.15]

(23.4) The Electric Field ( E )(23.4) The Electric Field ( E )

0

FEq

= Where q Where q 0 0 ≡≡ + test charge+ test charge

0

Fgm

=Compare with Gravitational Field:Compare with Gravitational Field:

EE

By convention: E points in By convention: E points in direction +q would movedirection +q would move

(23.6) Electric Field Lines(23.6) Electric Field Lines

Page 708, 9Page 708, 9thth ed.:ed.:

1.1. The lines begin on a positive charge and end on a The lines begin on a positive charge and end on a negative charge.negative charge.

2.2. Number of lines is proportional to magnitude of Q.Number of lines is proportional to magnitude of Q.3.3. No two field lines may cross.No two field lines may cross.

•• Also called Electric Also called Electric ““FluxFlux”” lines. lines.

(23.4) The Electric Field ( E )(23.4) The Electric Field ( E )

Check the units for Electric Field (E):Check the units for Electric Field (E):

0kq q

=2

0(rq )0q

=FE 2 ˆ

kq rr

=

r̂ ≡ unit vector

Problem [23.15]; find electric field at location of qProblem [23.15]; find electric field at location of q 77

(23.5) E of continuous charge (23.5) E of continuous charge distributiondistribution

2ˆkq r

r=E

And, of course, when q is NOT a point chargeAnd, of course, when q is NOT a point charge……..

2

kd dqr

=E 2k dqr

→ = ∫E


Problem [Ex: 23.7, page 705, 9Problem [Ex: 23.7, page 705, 9thth ed.]ed.]

(23.7) Motion of charges in Electric (23.7) Motion of charges in Electric FieldField

FEq

= F qE ma= =→

Problem [23.51]Problem [23.51]


QV

ρ =Charge distribution for volume:Charge distribution for volume:

QA

σ =Charge distribution for area:Charge distribution for area:

QL

λ =Charge distribution for length:Charge distribution for length:


Revised: 7/30/2013Revised: 7/30/2013

Electric FieldElectric Fieldat at thethe

Gas PumpGas Pump

linklink


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric Field The Electric Field


For highly symmetrical configurations of charge, Gauss For highly symmetrical configurations of charge, Gauss gave us a simple way of determining the electric field. It gave us a simple way of determining the electric field. It also serves as a guide to understanding complicated also serves as a guide to understanding complicated configurations of charge as well. configurations of charge as well.


Ch 24: GaussCh 24: Gauss’’s Laws Law

(24.1) Electric Flux(24.1) Electric Flux(24.2) Gauss(24.2) Gauss’’s Law s Law

(24.1) Electric Flux(24.1) Electric Flux

Electric FluxElectric Flux: # of electric field lines penetrating : # of electric field lines penetrating some surfacesome surface

Electric Flux E≡ Φ

E E A⊥→

Φ =2

2N NmmC C

→ =

E E A⊥→

Φ = E cosA θ=

(24.1) Electric Flux(24.1) Electric Flux

DotDot Product or Product or ScalarScalar Product:Product:

cosu v u v θ=i A scalar !A scalar !

EΦ = ∆∑ E Ai d= ∫ E AiOf Course, this is why we canOf Course, this is why we can’’t t

use use ““dotdot”” for multiply.for multiply.

(24.2) Gauss(24.2) Gauss’’s Laws Law

Just another method for calculating Electric FluxJust another method for calculating Electric Flux

GaussGauss’’s Laws Law: The net electric flux through ANY closed : The net electric flux through ANY closed (gaussian) surface is equal to the net charge inside the (gaussian) surface is equal to the net charge inside the surface divided by a constant.surface divided by a constant.

(24.2) Gauss(24.2) Gauss’’s Laws Law

E d⊥Φ = E AinsideQ

ε0= ∑

212

28.85(10 )C

Nmε −0 =



Revised: 7/30/2013Revised: 7/30/2013

Now, that shows how important it is to distinguish “fertilizing” practices from “fertility” practices when downloading a video file from the Internet.


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• GaussGauss’’s Law s Law


The electrostatic force is a conservative force and, The electrostatic force is a conservative force and, therefore, it is meaningful to talk about an electric therefore, it is meaningful to talk about an electric potential energy. This in turn enables us to define an potential energy. This in turn enables us to define an electrostatic potential, a concept simpler than force to electrostatic potential, a concept simpler than force to understand, because it is a scalar, in describing circuits understand, because it is a scalar, in describing circuits and many other electrical phenomena. and many other electrical phenomena.


Ch 25: Electric PotentialCh 25: Electric Potential

(25.1) Potential Difference and Electric Potential(25.1) Potential Difference and Electric Potential(25.2) Potential Differences in uniform E(25.2) Potential Differences in uniform E(25.3) Electric Potential & Potential Energy due to (25.3) Electric Potential & Potential Energy due to

pointpoint charges.charges.(25.5) When charge is not point charge(25.5) When charge is not point charge(25.7) Millikan (25.7) Millikan –– measuring charge of electronmeasuring charge of electron

(25.1) Potential Difference and (25.1) Potential Difference and Electric PotentialElectric Potential

Change in Potential Energy:Change in Potential Energy:

W = F sd d

0dU q E ds= −

0= ( ) sq E dThe Electric Field does work The Electric Field does work ONON q, so U of Electric q, so U of Electric Field Field decreasesdecreases::


For finite displacement of test charge between A, B:For finite displacement of test charge between A, B:

0

B

A

U q E ds∆ = − ∫ Path NOT important !Path NOT important !

Electric Field loses Electric Field loses energyenergy


Quote from your text:Quote from your text:The Potential Energy per unit charge ( U / qThe Potential Energy per unit charge ( U / q0 0 ) is ) is INdependentINdependent of the value of q of the value of q 00 and has a unique and has a unique value at EVERY point in the Electric Field.value at EVERY point in the Electric Field.

Quote from your text:Quote from your text:The Potential Energy per unit charge ( U / qThe Potential Energy per unit charge ( U / q0 0 ) is ) is INdependentINdependent of the value of q of the value of q 00 and has a unique and has a unique value at EVERY point in the Electric Field.value at EVERY point in the Electric Field.

Electric Potential V≡ ∆0

Uq∆

=

A scalar !A scalar !A scalar !A scalar !


Potential Difference Potential Difference ≡≡ V∆ =

0

B

A

U q E ds∆ = − ∫

0

Uq∆

Previously:Previously:

B

A

E ds= −∫

Potential Difference Potential Difference ≠≠ Potential EnergyPotential Energy


The Electric Potential (V) @ some arbitrary point (P) The Electric Potential (V) @ some arbitrary point (P) equals the work per charge done to bring a + test charge equals the work per charge done to bring a + test charge from infinity to that point.from infinity to that point.

JouleVoltCoulomb

→ =P

PV E ds∞

= −∫

0

UVq∆

∆ =B

A

E ds= −∫Previous slide:Previous slide:Does not indicate direction; Does not indicate direction;

V is a scalar.V is a scalar.

(25.1) Potential Difference and Electric (25.1) Potential Difference and Electric PotentialPotential

Another unit of energy:Another unit of energy:

191 1.6(10 )eV CV−= 191.6(10 ) J−=

Another unit for Electric Field (E):Another unit for Electric Field (E):

( )( )

N mC m

= 1JC m

⎛ ⎞= ⎜ ⎟⎝ ⎠

( )( )NmC m

= Vm

=NEC

⇒


A problem A problem notnot in your text:in your text:

A spark plug in your gasoline engineA spark plug in your gasoline engined d GapGap = 0.060 cm= 0.060 cmE = 3.0 (10E = 3.0 (1066) V/m) V/mCalculate Calculate ∆∆ V of the gap V of the gap

∆∆ V = E V = E ∆∆dd

(25.2) Potential Differences in (25.2) Potential Differences in uniformuniform EE

Again, work done in moving Again, work done in moving ““qq”” from A to B is from A to B is INDEPENDENTINDEPENDENT of path taken.of path taken.

B

A BA

V E ds Ed−∆ = − = −∫

(25.2) Potential Differences in (25.2) Potential Differences in uniformuniform EE

Equipotential Line:Equipotential Line:Any line consisting of a continuous distribution Any line consisting of a continuous distribution of points having the same Electric Potential (V).of points having the same Electric Potential (V).

Equipotential SurfaceEquipotential Surface: : Any surface consisting of a continuous Any surface consisting of a continuous distribution of points having the same Electric distribution of points having the same Electric Potential (V).Potential (V).


(25.3) Electric Potential & Potential (25.3) Electric Potential & Potential Energy Due to Energy Due to PointPoint ChargeCharge

0

UVq∆

∆ =

0U V q∆ = ∆

0F d Vq∆ = ∆

(25.3) Electric Potential & Potential Energy(25.3) Electric Potential & Potential Energy

From From Previous Previous Slide:Slide: 0

F d Vq∆ = ∆

02

k q q d Vqr

∆ = ∆

2

kq d Vr

∆ = ∆

(25.3) Electric Potential & Potential Energy(25.3) Electric Potential & Potential Energy

kq Vr

= ∆

ONLY for point ONLY for point charges !charges !


See note for this problem on next slideSee note for this problem on next slide

[25.16][25.16]

It takes 385 It takes 385 nJnJ of energy to push the charges apart. Iof energy to push the charges apart. I’’m doing work m doing work to push them apart and that work is given to push them apart and that work is given TOTO them in potential them in potential energy because they want to return toward each other and we can energy because they want to return toward each other and we can get that energy back as they move toward one another. get that energy back as they move toward one another.

However, if they were of like sign, I would do 385 However, if they were of like sign, I would do 385 nJnJ of work to push of work to push them together and that energy is given them together and that energy is given TOTO them in potential energy them in potential energy because they want to move away from each other and we can get because they want to move away from each other and we can get that energy back as they move away from one another.that energy back as they move away from one another.

So, the sign on the answer in part (a) is there due to the arbitSo, the sign on the answer in part (a) is there due to the arbitrary rary positive or negative sign that we have assigned to electrons andpositive or negative sign that we have assigned to electrons andprotons and not because of adding or removing energy to or from protons and not because of adding or removing energy to or from a a system.system.

(25.5) V due to continuous charge (25.5) V due to continuous charge distributiondistribution

Aka: V when charge is not a point chargeAka: V when charge is not a point charge

And, of courseAnd, of course……ekV dqr

= ∫

ek qVr

∆ = ekdV dqr

=→

Problem [25.45]Problem [25.45]Homework problem: Show sub from integral table Homework problem: Show sub from integral table

(25.7) The Millikan Oil(25.7) The Millikan Oil--Drop ExperimentDrop Experiment

Read about how Mr. Millikan measured the charge Read about how Mr. Millikan measured the charge of the electron.of the electron.

Our selection committee could not agree on who should get the last opening at the college. So, if you will just hold the spoon there for as long as you can…Martin –are you timing this?

(25.4) Obtain E from Electric Potential(25.4) Obtain E from Electric Potential

dV E ds= − iFrom From Earlier:Earlier:Now, if we only evaluate one component (one axis):Now, if we only evaluate one component (one axis):Partial derivative from Mechanics (Chapter 8)Partial derivative from Mechanics (Chapter 8)

x xE ds E d=i x xV E→∂ = − ∂

d∂ ≠ Partial derivativePartial derivative

(25.4) Obtain E from Electric Potential(25.4) Obtain E from Electric Potential

xVEx

∂= −

∂And,And,

yVEy

∂= −

∂ zVEz

∂= −

∂Problem [25.35; 8Problem [25.35; 8thth ed.] ed.]


Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• GaussGauss’’s Law s Law •• Electric Potential (V)Electric Potential (V)


Capacitors are devices used to store electrical charge. They Capacitors are devices used to store electrical charge. They usually involve conducting plates separated by an usually involve conducting plates separated by an insulator.insulator.


Ch 26: Capacitance & DielectricsCh 26: Capacitance & Dielectrics

(26.1) Definition of Capacitance(26.1) Definition of Capacitance(26.2) Calculating Capacitance(26.2) Calculating Capacitance(26.3) Capacitors in series & parallel (26.3) Capacitors in series & parallel (26.4) Energy stored in a capacitor(26.4) Energy stored in a capacitor(26.5) Capacitor with dielectric(26.5) Capacitor with dielectric(26.6) Electric dipole in an Electric Field (26.6) Electric dipole in an Electric Field

(26.1) Definition of Capacitance(26.1) Definition of Capacitance

Capacitors store charge.Capacitors store charge.

QCV

≡ Where Q Where Q ≡≡ charge on charge on EITHEREITHER plateplate

1 Coulomb1 Farad 1 Volt

≡

(26.1) Definition of Capacitance(26.1) Definition of Capacitance

Demonstration of charging & discharging a capacitorDemonstration of charging & discharging a capacitor


Now, skip to Section (26.5)Now, skip to Section (26.5)

(26.5) Capacitor with dielectric(26.5) Capacitor with dielectric

0ACd

κ ε=Where:Where:k k ≡≡ dielectric constant (next slide)dielectric constant (next slide)A A ≡≡ areaaread d ≡≡ distance between platesdistance between plates

2-12

o 2

C = 8.85(10)Nm

ε

ProblemProblem[26.6][26.6]

(26.3) Capacitors in series & parallel(26.3) Capacitors in series & parallel

No potential No potential loss in perfect loss in perfect wireswires

Capacitors in Parallel:Capacitors in Parallel:

1 2 3 ...A BC C C C− = + + +

AA

BB

CC11 CC22 CC33

1 2 3 ...V V V= = =

1 2 3 ...A BQ Q Q Q− = + + +

(26.3) Capacitors in series & parallel(26.3) Capacitors in series & parallel

Capacitors in Series:Capacitors in Series:

1 2 3

1 1 1 1 ...A BC C C C−

= + + +

1 2 3 ...A BV V V V− = + + +

AA

BB

CC11

CC22

CC33

Careful !Careful !

1 2 3 ...A BQ Q Q Q− = = = = Problem [26.23] Problem [26.23]

(26.4) Energy stored in a capacitor(26.4) Energy stored in a capacitor

Compare with previous expressions for Energy:Compare with previous expressions for Energy:

2

2QUC

= 12 QV=21

2 CV=

212TranslationalK mv= 21

2springU k x=21

2RotationalK Iω= All are physical All are physical descriptionsdescriptions

(26.4) Energy stored in a capacitor(26.4) Energy stored in a capacitor

Now, back to Problem [26.6]Now, back to Problem [26.6]Find energy stored by Find energy stored by ““cloud capacitorcloud capacitor””

t t releaserelease = 0.033 s= 0.033 s

TVA record as of 7TVA record as of 7--1818--06: 32 BW06: 32 BWFind value if rate isFind value if rate is 8.75 cent

kWHr

(26.6) Electric dipole in an Electric Field(26.6) Electric dipole in an Electric Field

Electric Dipole :Electric Dipole :

p X Eτ =

Finish capacitor demonstrationFinish capacitor demonstration

1


Revised: 7/30/2013Revised: 7/30/2013 Circuits Lab


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• CapacitanceCapacitance


Until now, we have considered only electrostatics Until now, we have considered only electrostatics –– that is, that is, charges at rest. We now want to consider electrical charges at rest. We now want to consider electrical current, which is the time rate of flow of charges in current, which is the time rate of flow of charges in motion. Also, of interest, is current density which is the motion. Also, of interest, is current density which is the amount of current per unit area. We will also consider the amount of current per unit area. We will also consider the factors that enhance or inhibit charge flow which is the factors that enhance or inhibit charge flow which is the study of electrical resistance. study of electrical resistance.


Ch 27: Current & ResistanceCh 27: Current & Resistance

(27.1) Electric Current(27.1) Electric Current(27.2) Resistance & Ohm(27.2) Resistance & Ohm’’s Laws Law(27.3) A Model for electrical Conduction(27.3) A Model for electrical Conduction(27.4) Resistance and Temperature(27.4) Resistance and Temperature(27.5) Superconductors(27.5) Superconductors(27.6) Electrical Energy and Power (27.6) Electrical Energy and Power

(27.1) Electric Current ( I )(27.1) Electric Current ( I )

dQIdt

≡Where:Where:n = charge carrier / volumen = charge carrier / volumeq = q = charge of each carriercharge of each carrierv v dd = drift velocity= drift velocityA = cross sectional areaA = cross sectional area

1 Coulomb1 Ampere = 1 second

dn q v A=

Units:Units:

I I ≡≡ electrical currentelectrical current

2

(27.1) Electric Current ( I )(27.1) Electric Current ( I )

Problem:Problem:A wire, 1.00A wire, 1.00--m in length and 1.29 mm in diameter, carries m in length and 1.29 mm in diameter, carries a steady current of 1.00 A. If the conductor is copper a steady current of 1.00 A. If the conductor is copper which has a free charge density of 8.5 (10which has a free charge density of 8.5 (102828) electrons per ) electrons per cubit meter, what is the velocity of one electron? (Direct cubit meter, what is the velocity of one electron? (Direct current is assumed)current is assumed)

(27.2) Resistance & Ohm(27.2) Resistance & Ohm’’s Laws Law

OhmOhm’’s Law:s Law:

Current Density J≡ CurrentArea

≡ IA

=

JE

σ =Where:Where:σσ ≡≡ conductivityconductivity


( )Volt Amp Ohm→ =

VI

=

resistivityρ ≡ 1σ

=

Another version of OhmAnother version of Ohm’’s Law:s Law: V I R=Units:Units: V A= Ω

LRAσ

=


LRA

ρ∴ =

(27.3) A Model for electrical (27.3) A Model for electrical ConductionConduction

Random motion of electronsRandom motion of electrons

electronsWithout , 0E vΣ =

4driftWith , (10)

mE vs

−≈

(27.4) Resistance and Temperature(27.4) Resistance and Temperature

Where:Where:∝∝ ≡≡ Temperature Temperature coefficient of coefficient of

resistivityresistivityρρ ≡≡ ResistivityResistivity

T T ≡≡ Temperature Temperature

0 Tρ ρ α∆ = ∆

3

(27.4) Resistance and Temperature(27.4) Resistance and Temperature

0 Tρ ρ α∆ = ∆


And, because :And, because : 0R R Tα→∆ = ∆R ρ∝

(27.5) Superconductors(27.5) Superconductors

SUPERSUPERconductorconductor: A really good conductor: A really good conductorSo good, in fact:So good, in fact:

0R =

Be sure to read this section.Be sure to read this section.

(27.6) Electrical Energy and Power(27.6) Electrical Energy and Power

As charge moves from As charge moves from ““AA”” to to ““BB””, Electric Potential , Electric Potential Energy INCREASES by [Energy INCREASES by [∆∆V Q] and chemical Potential V Q] and chemical Potential Energy of the battery DECREASES by [Energy of the battery DECREASES by [∆∆V Q]. V Q].

AA

BBRR∆∆VV

--

++

DD

CCII

Figure from your text:Figure from your text:

UVq

∆∆ =

∆

( )U V q∆ = ∆ ∆


AA

BBRR∆∆VV

--

++

DD

CCII

As the charge moves C to D, it losses this Electrical As the charge moves C to D, it losses this Electrical Energy (collisions with atoms in resistor) producing Energy (collisions with atoms in resistor) producing thermal energy. thermal energy.

No loss in wire (for a No loss in wire (for a perfectperfect conductor). No conductor). No ∆∆V in wire.V in wire.So, So, ““DD”” is at 0 Volt as is is at 0 Volt as is ““AA””..

( )U V q∆ = ∆ ∆


Power: Rate of work or Rate of using energy.Power: Rate of work or Rate of using energy.

Rate of loss of Potential Energy in Resistance:Rate of loss of Potential Energy in Resistance:

UVQ

∆=∆

U QV→∆ = ∆

U Q Vt t

∆ ∆⎡ ⎤= = ⎢ ⎥∆ ∆⎣ ⎦I V P= =WorkP

time≡


( )P I I R=


Other forms of :Other forms of :

2VPR

=

P I V=V I R=

VIR

=

2P I R=

P I V=VP VR⎡ ⎤= ⎢ ⎥⎣ ⎦


Revised: 7/30/2013Revised: 7/30/2013

Physics Lab


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• CapacitanceCapacitance•• Current & Resistance Current & Resistance


In this chapter, we study the rules that govern how In this chapter, we study the rules that govern how charges move in circuits which have resistance and charges move in circuits which have resistance and multiple branches. We will take into account what we multiple branches. We will take into account what we know about direct current processes and sources of know about direct current processes and sources of electrical potential. electrical potential.


Ch 28: Direct Current CircuitsCh 28: Direct Current Circuits

(28.1) Electromotive Force ((28.1) Electromotive Force (E))(28.2) Resistors in Series & Parallel(28.2) Resistors in Series & Parallel(28.3) Kirchhoff(28.3) Kirchhoff’’s Ruless Rules(28.4) RC Circuits(28.4) RC Circuits(28.6) Household wiring and Electrical Safety (28.6) Household wiring and Electrical Safety

(28.1) Electromotive Force(28.1) Electromotive Force

From your text:From your text:

r r ≡≡ internal resistanceinternal resistanceεε ≡≡ emf (emf (poor termpoor term))R R ≡≡ Load ResistorLoad Resistor

A positive charge moves A positive charge moves ‘‘aa’’ to to ‘‘bb’’. As +q passes . As +q passes through through εε, its electric potential (V) , its electric potential (V) increasesincreases by by εε. .

But, as +q moves through r, V But, as +q moves through r, V decreasesdecreases by I r.by I r.Then, as +q moves through R, V Then, as +q moves through R, V decreasesdecreases by I R.by I R.

a b

cd R

r intE

- +

(28.1) Electromotive Force(28.1) Electromotive Force

Terminal inta bV V I rε− = = −

→→ r r intint can be neglected if R >> r can be neglected if R >> r intint


Terminal RV V I R= =

intI R I rε = + int( )I R r

a b

cd R

r intE

- +

= +

int( )I

R rε

=+

(28.2) Resistors in Series & Parallel(28.2) Resistors in Series & Parallel

RR11

VVTotalTotal

-- ++

RR22 RR33

IITTIITT

II11 II22 II33

SeriesSeries

1 2 3 TotalI = I = I = I i i i

1 2 3 TotalV + V + V + = V i i i

Equivalent 1 2 3 TotalR = R + R + R + = R i i i


ParallelParallel

1 2 3 TotalI I I I+ + + =i i i

VV

--

++

RR33RR22RR11 II22

II TotalTotal

II TotalTotal

II11 II33

1 2 3 TotalV V V V= = =i i i

. 1 2 3

1 1 1 1

equivR R R R= + + + i i i Problem [28.18]Problem [28.18]


VV

--

++

RR33RR22RR11 II22

II TotalTotal

II TotalTotal

II11 II33


(28.3) Kirchhoff(28.3) Kirchhoff’’s Ruless RulesThe The sumsum of currents entering any junction = of currents entering any junction = sumsum of of currents leaving that same junction.currents leaving that same junction.

I I 11 I I 22

I I 33

1 2 3I I I= +

The algebraic The algebraic sumsum of of ∆∆ V across all elements around V across all elements around any any CLOSEDCLOSED circuit loop = 0circuit loop = 0

R1

VV

- +

R2 R3

(28.3) Kirchhoff(28.3) Kirchhoff’’s Ruless Rules

See aids in textSee aids in text _+ →→If we assume this If we assume this

direction for direction for currentcurrent……

→→Move this dir.Move this dir.

+ _

--V = V = --I RI R←←Move this dir.Move this dir.

+

+V = I R+V = I R

_

-+

+V+V

←←

……we get these we get these signssigns

As we sum around the loop, the As we sum around the loop, the direction we go determines the direction we go determines the

sign of V.sign of V.

-+

→→

--VVExample Problem; Example Problem; 28.7, pg 84628.7, pg 846

(28.3) (28.3) KirchhoffKirchhoff’’ss RulesRules

Your homework problem # 28:Your homework problem # 28:

The The ““dead batterydead battery”” should have a potential of 10 V, and should have a potential of 10 V, and not 12 V.not 12 V.

(28.4) RC Circuits(28.4) RC Circuits

Using a DC potential to charge a capacitor through a Using a DC potential to charge a capacitor through a resistor.resistor.

VVoo = + 10 V= + 10 V

RR

CC

Characteristic time constant:Characteristic time constant:

RCτ =

sec F= ΩUnits:Units:


VVoo = + 10 V= + 10 V

R = 100 k R = 100 k ΩΩ

C = 100 C = 100 µµFF

0( ) (1 )t

CapV t V eτ−⎡ ⎤⎢ ⎥⎣ ⎦= −

VoltmeterVoltmeter

CharginCharging:g:

1 TC

2 TC

3 TC

Charging:Charging:


VVoo = + 10 V= + 10 V

R = 100 k R = 100 k ΩΩ

C = 100 C = 100 µµFF

Discharging:Discharging:

0( )t

CapV t V eτ−⎡ ⎤⎢ ⎥⎣ ⎦=

(28.6) Household wiring and Electrical (28.6) Household wiring and Electrical SafetySafety

You should You should readread this section carefully.this section carefully.

Topics:Topics:

1.1. OpenOpen2.2. ShortShort3.3. Series vs. ParallelSeries vs. Parallel4.4. G.F.C.I. (Ground Fault Circuit Interrupter) G.F.C.I. (Ground Fault Circuit Interrupter)

(28.6) Household wiring and Electrical (28.6) Household wiring and Electrical SafetySafety

G.F.C.I. (Ground Fault Circuit Interrupter)G.F.C.I. (Ground Fault Circuit Interrupter)

TransformerTransformer PicturesPictures


Revised: 7/30/2013Revised: 7/30/2013

Kirchhoff’s Laws


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• CapacitanceCapacitance•• Current & Resistance Current & Resistance •• DC CircuitsDC Circuits


Magnetism is known to be connected with moving charges Magnetism is known to be connected with moving charges and is, therefore, closely connected to electricity. There are and is, therefore, closely connected to electricity. There are many devices using this phenomenon. In a way, magnets many devices using this phenomenon. In a way, magnets are devices which harness electricity in a package. Similar are devices which harness electricity in a package. Similar to electric charges, like magnetic poles repel and unlike to electric charges, like magnetic poles repel and unlike poles attract. A major difference, however, is that poles attract. A major difference, however, is that magnetic monopoles are not known to exist.magnetic monopoles are not known to exist.


Ch 29: The Magnetic FieldCh 29: The Magnetic Field

(29.1) The Magnetic Field(29.1) The Magnetic Field(29.4) Magnetic Force on a I(29.4) Magnetic Force on a I--carrying Conductorcarrying Conductor(23.5) Torque on Current loop in Magnetic Field (23.5) Torque on Current loop in Magnetic Field

(29.1) The Magnetic Field(29.1) The Magnetic Field

•• Magnetic Field (B) surrounds any magnetic substance Magnetic Field (B) surrounds any magnetic substance and, also, any and, also, any MOVINGMOVING charge.charge.

•• To date, there are To date, there are NONO known magnetic known magnetic MONOpolesMONOpoles..

•• Chapter 29 looks at forces on Chapter 29 looks at forces on MOVINGMOVING charges and charges and on currenton current--carrying wires in the presence of B.carrying wires in the presence of B.

•• Chapter 30 discusses Chapter 30 discusses SOURCESSOURCES of B.of B.


Compare different types of fields:Compare different types of fields:

0

Fgm

=Gravity Field:Gravity Field:

Electric Field:Electric Field:0

FEq

=

Symbol for Gravitational Field


For + qFor + qF qv X B=We first saw the cross We first saw the cross product in angular product in angular momentum, Ch. 11momentum, Ch. 11

F qv B⊥= sinqv B θ= θθ is between v and Bis between v and B

If If ⊥⊥, sin 90 , sin 90 00 = 1= 1

If If ⎮⎮⎮⎮, sin 0 = 0, so F = 0, sin 0 = 0, so F = 0


FF

vv

BB


SI units of the Magnetic Field (B):SI units of the Magnetic Field (B):

2 2

( ) ( )Weber WbB Tesla Tm m

≡ = ≡ N Nm AmCs

⇒ =

41 [10 ]T Gauss=NonNon--SI Units:SI Units: Gauss (G)Gauss (G)


(29.2) Magnetic Force on a (29.2) Magnetic Force on a CurrentCurrent--carrying Conductorcarrying Conductor

Same as F = q v X B only this is a collection of Same as F = q v X B only this is a collection of charges.charges.

Now, force on a (straight) INow, force on a (straight) I--carrying wire in presence carrying wire in presence of uniform B:of uniform B:

F I L X B=Where:Where: L L ≡≡ length of wirelength of wire

I I ≡≡ current in wirecurrent in wire

m CC ms s

=Units of q(v) vs. I(L) :Units of q(v) vs. I(L) :


FF FF


And, of course, for a wire with arbitrary shape:And, of course, for a wire with arbitrary shape:

dF I dL X B=b

a

F I dL X B⇒ = ∫


(29.3) Torque on Current loop in (29.3) Torque on Current loop in Magnetic FieldMagnetic Field

Compare with moment of inertia:Compare with moment of inertia:

Magnetic Moment Magnetic Moment ≡≡ 2I A Amp mµ = ⇒

2I mr=


I Aµ =

Same direction as Same direction as areaarea


X Bτ µ= sinBµ θ= sinN Bµ θ⇒For a single loopFor a single loop

For multiple loops For multiple loops (turns):(turns):

Compare with electrical torque on electric dipole:Compare with electrical torque on electric dipole:

p X Eτ =Problem [29Problem [29--48]48]Use v X B to determine direction of rotation.Use v X B to determine direction of rotation.


Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• CapacitanceCapacitance•• Current & Resistance Current & Resistance •• DC Circuits DC Circuits •• Magnetic FieldMagnetic Field


Because currents produce magnetic fields, we expect Because currents produce magnetic fields, we expect them to interact with one another through these fields. them to interact with one another through these fields. Practical devices can be constructed from these Practical devices can be constructed from these interactions. Some of the possibilities are discussed in interactions. Some of the possibilities are discussed in this chapter. this chapter.

This chapter looks at current in a wire as a source of This chapter looks at current in a wire as a source of Magnetic Field (B)Magnetic Field (B)

Section 8 discusses Section 8 discusses ““Magnetism in MatterMagnetism in Matter””..Be sure to read Section 9 on Be sure to read Section 9 on ““BB”” of the Earth.of the Earth.


Ch 30: Sources of The Magnetic FieldCh 30: Sources of The Magnetic Field

(30.4) Magnetic Field of Solenoid(30.4) Magnetic Field of Solenoid(30.5) Gauss(30.5) Gauss’’s Law in Magnetism (Magnetic Flux )s Law in Magnetism (Magnetic Flux )

(30.4) Magnetic Field of Solenoid(30.4) Magnetic Field of Solenoid

0SOLNB IL

µ=N = # of turnsN = # of turnsL = Length of solenoidL = Length of solenoidI = Current in solenoidI = Current in solenoid

70 4 (10)

TmA

µ π −=

(30.5) Magnetic Flux(30.5) Magnetic Flux

B B dAφ = ∫ i cosBA θ=2

B Tmφ ⇒Where:Where:

cosE E Aφ θ=Compare:Compare:


(30.6) Gauss(30.6) Gauss’’s Law in Magnetisms Law in Magnetism

0B dA =∫ iThe The netnet magnetic flux through any magnetic flux through any closedclosed surface is surface is always zero.always zero.

MagnetMagnetOutOut

OutOutInIn

InIn

Closed Closed SurfaceSurface

(30.4) Magnetic Field of Solenoid(30.4) Magnetic Field of Solenoid

0SOLNB IL

µ=N = # of turnsN = # of turnsL = Length of solenoidL = Length of solenoidI = Current in solenoidI = Current in solenoid

70 4 (10)

TmA

µ π −=

Use this slide if there is time for the following problem:


(30.1) The (30.1) The BiotBiot--SavartSavart LawLaw

If a wire carries a STEADY current, the B at point If a wire carries a STEADY current, the B at point ‘‘pp’’associated with wire element associated with wire element ‘‘dsds’’ has these properties:has these properties:

(30.1) The (30.1) The BiotBiot--SavartSavart LawLaw

••The vector, dB, is The vector, dB, is ⊥⊥ both to both to dsds (in direction of I) and r (in direction of I) and r (directed from (directed from dsds to to ‘‘pp’’))

••The magnitude of dB is The magnitude of dB is ∝∝ 1 / r1 / r22, where r is dist , where r is dist dsds to to ‘‘pp’’..

••The magnitude of dB is The magnitude of dB is ∝∝ I and magnitude of length I and magnitude of length element element dsds..

••The magnitude of dB is The magnitude of dB is ∝∝ sin sin θθ, where , where θθ is angle is angle between between ‘‘dsds’’ and and ‘‘rr’’..

••See Figure 30.1, page 927See Figure 30.1, page 927

(30.1) The (30.1) The BiotBiot--SavartSavart Law, contLaw, cont’’dd

710mT mkA

−=^

2mIdsdB k X rr

=

Example 30.1, page 929:Example 30.1, page 929:Find B for long, straight wire:Find B for long, straight wire: 0

2IBr

µπ

=

Example 30.2 page 930: B due to I in circular loop Example 30.2 page 930: B due to I in circular loop

(30.2) (30.2) FFmagmag between two parallel between two parallel conductorsconductors

Each conductor produces a magnetic fieldEach conductor produces a magnetic field

F I L X B=From From chch 29:29:

0 1 2

2I IF

L aµπ

=Where:Where:a = distance between wires

70 4 (10)

TmA

µ π −=

Now:Now:

I in same direction: I in same direction: attractattractI in opposite direction: I in opposite direction: repelrepel

(30.3) Ampere(30.3) Ampere’’s Laws Law

Valid only for Valid only for steadysteady currentscurrentsUseful Useful onlyonly when current configuration is highly when current configuration is highly symmetrical.symmetrical.

0B ds Iµ=∫ iCompare Ex: 30.4, page 935 with Ex: 30.1Compare Ex: 30.4, page 935 with Ex: 30.1


Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• CapacitanceCapacitance•• Current & Resistance Current & Resistance •• DC Circuits DC Circuits •• Magnetic FieldMagnetic Field•• GaussGauss’’s Laws Law


Magnetic fields not only exert a force on moving charge, Magnetic fields not only exert a force on moving charge, but also a timebut also a time--varying magnetic field produces an electric varying magnetic field produces an electric field. Faraday's law of induction relates magnetic flux field. Faraday's law of induction relates magnetic flux change to the electromotive force produced, a process change to the electromotive force produced, a process called induction. called induction. Inductors are circuit elements that produce an emf that Inductors are circuit elements that produce an emf that opposes an external emf that is causing a change to occur. opposes an external emf that is causing a change to occur. This is an effect known as Lenz's law and leads to a This is an effect known as Lenz's law and leads to a process called selfprocess called self--induction. induction.


Ch 31: FaradayCh 31: Faraday’’s Laws Law

(31.1) Faraday(31.1) Faraday’’s Law of Inductions Law of Induction(31.2) Motional emf(31.2) Motional emf(31.3) Lenz(31.3) Lenz’’s Laws Law(31.4) Induced emf and Electric Fields(31.4) Induced emf and Electric Fields(31.5) Generators & Motors(31.5) Generators & Motors

(31.1) Faraday(31.1) Faraday’’s Law of Inductions Law of Induction

From earlier:From earlier:E due to static chargeE due to static chargeB due to moving chargesB due to moving charges

Now:Now: E as result of changing B E as result of changing B ⇒⇒ InductionInduction


BdNdtφε = −

cosB BAφ θ=

•• N N ≡≡ # of loops (turns)# of loops (turns)•• Negative sign laterNegative sign later……


•• Emf (Emf (εε) can be induced several ways:) can be induced several ways:

•• Change Change ⎮⎮BB⎮⎮ with timewith time•• Change A (area of loop) with timeChange A (area of loop) with time•• Change Change θθ with time (angle between A and B)with time (angle between A and B)•• Change N (number of loops)Change N (number of loops)•• Any combination of the aboveAny combination of the above

Magnitude of induced emf depends upon Magnitude of induced emf depends upon rate of rate of changechange of magnetic flux.of magnetic flux.


(31.2) Motional emf(31.2) Motional emf

Emf induced in wire due to moving through BEmf induced in wire due to moving through B

Wire WireB L vε ⊥= −

Negative sign nextNegative sign next……A problem laterA problem later……

(31.3) Lenz(31.3) Lenz’’s Laws Law

BdNdtφε = −

Negative sign: Negative sign: The polarity of induced emf produces a current which The polarity of induced emf produces a current which creates flux to oppose original flux.creates flux to oppose original flux.

Demonstration of LenzDemonstration of Lenz’’s Laws Law


(31.5) Generators & Motors(31.5) Generators & Motors

Read this section carefully.Read this section carefully.

Java Demonstration Java Demonstration of ac alternatorof ac alternator

It is the It is the changechange in flux, not just flux.in flux, not just flux.

εε maxmax : : εε (t) when sides of loop are parallel to B(t) when sides of loop are parallel to B

use: F = q v B sin use: F = q v B sin θθ

(31.5) Generators & Motors(31.5) Generators & Motors

( ) sint NAB tε ω ω=

max NABε ω= Where sin Where sin ωωt = (+/t = (+/--) 1) 1


(31.4) Induced emf ( (31.4) Induced emf ( εε ) ) and Electric Field (E)and Electric Field (E)

BdE dsdtφ

= −∫ iAn E is created in the conductor as a result of An E is created in the conductor as a result of ∆∆ φφBB ..

⇒⇒ Caused by change in Caused by change in φφBB

0A A AV V V E ds∆ = − = − =∫ i

↑↑ NonNon--ConservativeConservative

↑↑ ConservativeConservative

Above equation Above equation cannotcannot be electrostatic (conservative)be electrostatic (conservative)because from Ch. 25:because from Ch. 25: From A to AFrom A to A

(31.4) Induced emf ((31.4) Induced emf (ε ) ) and Electric Field (E)and Electric Field (E)

So, E due to static charge is So, E due to static charge is differentdifferent from E due to from E due to changing magnetic flux.changing magnetic flux.

If E were electrostatic (If E were electrostatic (conservativeconservative), the line ), the line integral of E integral of E •• dsds over a closed loop would = 0over a closed loop would = 0


Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• CapacitanceCapacitance•• Current & Resistance Current & Resistance •• DC Circuits DC Circuits •• Magnetic FieldMagnetic Field•• FaradayFaraday’’s Laws Law


Because a changing current produces a changing magnetic Because a changing current produces a changing magnetic flux, an emf is set up which opposes the change just as flux, an emf is set up which opposes the change just as stated in Lenz's law. The proportionality constant between stated in Lenz's law. The proportionality constant between this induced backthis induced back--emf that opposes the changing current emf that opposes the changing current and the changing magnetic flux is called inductance. We and the changing magnetic flux is called inductance. We establish selfestablish self--inductance when the coil or circuit element inductance when the coil or circuit element acts on itself and mutual inductance when one part of the acts on itself and mutual inductance when one part of the circuit interacts with another part, like one coil interacting circuit interacts with another part, like one coil interacting with another coil. with another coil.


Ch 32: InductanceCh 32: Inductance

(32.1) Self(32.1) Self--induction and Inductanceinduction and Inductance(32.2) RL Circuits(32.2) RL Circuits(32.3) Energy in Magnetic Field(32.3) Energy in Magnetic Field(32.4) Mutual Inductance(32.4) Mutual Inductance(32.5) Oscillations in LC Circuit (32.5) Oscillations in LC Circuit

(32.1) Self(32.1) Self--inductanceinductance

CoilCoil+V+VDCDC

SwitchSwitch

ResistorResistor

What happens when the Switch is closed?What happens when the Switch is closed?•• Counter emf (back emf)Counter emf (back emf)•• LenzLenz’’s Laws Law

Notice DC Notice DC voltagevoltage


The selfThe self--induced emf (counterinduced emf (counter--emf) is always emf) is always proportional to:proportional to:

dIdt

ε ∝


For closelyFor closely--spaced coil of N turns (i. e., ideal solenoid):spaced coil of N turns (i. e., ideal solenoid):

BL

dNdtφε = −

FaradayFaraday’’s Laws LawdILdt

= −

Where Where εεLL ≡≡ selfself--induced (counter) emfinduced (counter) emfL L ≡≡ inductance (proportionality constant)inductance (proportionality constant)

(32.1) Self(32.1) Self--inductanceinductanceUnits of Induction:Units of Induction:

secVoltHenryAmp

≡ sVHA

≡

Example 32.1, page 929Example 32.1, page 929 Inductance of Solenoid:Inductance of Solenoid:

2

oN AL

lµ=


(32.2) RL Circuits(32.2) RL CircuitsCharacteristic time of a Characteristic time of a seriesseries resistor and inductor:resistor and inductor:

LR

τ = sec HenryOhm

=

Current (I) will reach equilibrium value Current (I) will reach equilibrium value after a time which is after a time which is longlong compared to compared to ττ..

⇒

MAXI Rε

=

VV

a

b

RE

-

+L

S

+

-

I

Inductor has no effect at this time.Inductor has no effect at this time.

(32.2) RL Circuits(32.2) RL Circuits

Compare:Compare:

I vs. time curve for RL circuitI vs. time curve for RL circuit

v.v.

V vs. time curve for RC circuitV vs. time curve for RC circuit

Next SlideNext Slide

RC CircuitRC Circuit

Charging:Charging:

1 TC

2 TC

3 TC

TimeTime

VVCapCap

Charging:Charging:

1 TC

2 TC

3 TC

RL CircuitRL Circuit

TimeTime

IICoilCoil


( ) [1 ]t

I t eR

τε−⎧ ⎫

⎨ ⎬⎩ ⎭= −

For increasing current:For increasing current:

FinalIRε=


And for decreasing current:And for decreasing current: ( ) [ ]t

I t eR

τε−⎧ ⎫

⎨ ⎬⎩ ⎭=


(32.3) Energy in Magnetic Field(32.3) Energy in Magnetic Field

Power Power ↓↓

Rate of heat Rate of heat ↑↑↑↑Rate of E stored in BRate of E stored in B

Rate of Energy Power Iε≡ =

212LU L I= Problem [32.34]Problem [32.34]

2I I Rε =dIL Idt

⎡ ⎤+ ⎢ ⎥⎣ ⎦

VV

(32.4) Mutual Inductance(32.4) Mutual Inductance

Interaction between two circuits (coils)Interaction between two circuits (coils)

Good example: TransformerGood example: Transformer

(32.5) Oscillations in LC Circuit(32.5) Oscillations in LC Circuit

Total energy in LC circuit remains constant so,Total energy in LC circuit remains constant so,……

0dUdt

= But with Resistance But with Resistance ……

(32.5) Oscillations in LC Circuit(32.5) Oscillations in LC Circuit

Total energy in LC circuit remains constant so,Total energy in LC circuit remains constant so,……

0dUdt

=

Total Cap InductorU U U= +

But with Resistance But with Resistance ……

2 22 2cos sin

2 2Max MaxQ LIt tC

ω ω= +

↑↑ 180 180 oo out of phase out of phase ↑↑


Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• DC Circuits DC Circuits •• Magnetic FieldMagnetic Field•• InductionInduction•• Inductance Inductance


Alternating Currents (ac) are found in our homes and in Alternating Currents (ac) are found in our homes and in our work places and have become a part of our everyday our work places and have become a part of our everyday life. We need to understand the basic physics associated life. We need to understand the basic physics associated with them. To do this we need to know how resistors, with them. To do this we need to know how resistors, capacitors, and inductors work in ac circuits. capacitors, and inductors work in ac circuits.


Ch 33: AlternatingCh 33: Alternating--Current CircuitsCurrent Circuits(33.1) AC Sources (33.1) AC Sources (33.2) Resistors in ac Circuits(33.2) Resistors in ac Circuits(33.3) Inductors in ac Circuits(33.3) Inductors in ac Circuits(33.4) Capacitors in ac Circuits(33.4) Capacitors in ac Circuits(33.5) The RLC Series Circuit(33.5) The RLC Series Circuit(33.6) Power in ac Circuit(33.6) Power in ac Circuit(33.8) The Transformer & Power Transmission(33.8) The Transformer & Power Transmission

(33.1) ac Sources and (33.1) ac Sources and PhasorsPhasors[Rotating Vectors][Rotating Vectors]

For sinusoidal waveforms (such as our ac from TVA):For sinusoidal waveforms (such as our ac from TVA):

instantaneous ( ) sinMaxV v t V tω≡ =

And, of course:And, of course: 2 fω π=1fT

=

i, v i, v ≡≡ instantaneous instantaneous values.values.

(33.2) Resistors in ac Circuits(33.2) Resistors in ac Circuits

Also, Also,

( ) sinMaxi t I tω=

( ) sinMaxv t V tω=Now, if:Now, if:

Then:Then: ( ) ( )sinMaxv t I R tω=

(33.2) Resistors in ac Circuits(33.2) Resistors in ac Circuits

•• RMS RMS ≡≡ rootroot--meanmean--squaresquare•• The square root of the mean value of ( The square root of the mean value of ( ∑∑ I ) I ) 22

2Max

RMS effectiveII I≡ =

And,And,

Problem [33.2]Problem [33.2]2

MaxRMS effective

VV V≡ =

(33.3) Inductors in ac Circuits(33.3) Inductors in ac Circuits

What happens when the switch closes?What happens when the switch closes?

Inductive Reactance:Inductive Reactance: 2LX f L Lπ ω= =

CoilCoil+V+VDCDC

SwitchSwitch

ResistorResistor

V leads I by 90 V leads I by 90 o o in in purelypurely inductive circuit.inductive circuit.

⇒ΩNotice DCNotice DC

JavaJava““UU”” is is ““VV””

(33.4) Capacitors in ac Circuits(33.4) Capacitors in ac Circuits

What happens when the switch closes?What happens when the switch closes?

Capacitive Reactance:Capacitive Reactance:1 1

2CX

f C Cπ ω= =

I leads V by 90 I leads V by 90 o o in in purelypurely capacitive circuit.capacitive circuit.

⇒Ω

+V+VDCDC

SwitchSwitch

ResistorResistor CapacitorCapacitor

JavaJava““UU”” is is ““VV””

(33.4) Capacitors in ac Circuits(33.4) Capacitors in ac Circuits

II CC EE EE LL II

(33.5) The RLC Circuit(33.5) The RLC Circuit

RR

XXCC

XXLL

+y+y

+x+x

--yy

~~RR

XXCC

XXLL

Problem [33.29]Problem [33.29] Graph on next slide Graph on next slide →→

(33.5) The RLC Circuit(33.5) The RLC Circuit

-250.00

-200.00

-150.00

-100.00

-50.00

0.00

50.00

100.00

150.00

200.00

250.00

1 15 29 43 57 71 85 99 113 127 141 155 169 183 197

V(t)VLVC VR

Excel SpreadsheetExcel Spreadsheet

(33.6) Power in ac Circuit(33.6) Power in ac Circuit

NoNo power loss (energy used) in power loss (energy used) in purepure inductor or inductor or purepurecapacitor.capacitor.

.instP i v= More on this next slideMore on this next slide

cosRMS RMSP I V θ=

where where coscos θθ ≡≡ power factor or p.f.power factor or p.f.


θθ

VVRR

VVLL--VVCCVVAppliedApplied

θθ

RR

XXLL--XXCCZZ

cos adj Rhyp Z

θ = = cosRApplied

VV

θ⇒ =OROR

( . .)AppliedV p f=cosR AppliedV V θ=


Power used Power used onlyonly by Resistance:by Resistance:

From previous slideFrom previous slide[ ]RP I V=

[ cos ]R AppliedI V θ= ( . .)AppliedI V p f=


2Real Power. .Apparent Power

I Rp fV A

= =

2average RMSP I R=But, Always:But, Always:


(33.8) The Transformer & Power (33.8) The Transformer & Power TransmissionTransmission

primary secondaryPower = Power

Primary Primary coilcoil

Secondary Secondary coilcoil

Also, approximately:Also, approximately:sec sec

pri priV NV N

=Loss?Loss?

(33.7) Resonance in series RLC Circuit(33.7) Resonance in series RLC Circuit

Occurs when XOccurs when XLL = X= XCC

Demo Demo of RLC circuit resonance of RLC circuit resonance



Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• DC Circuits DC Circuits •• Magnetic FieldMagnetic Field•• InductionInduction•• Inductance Inductance •• ac circuits ac circuits


Motion that repeats itself is very common and applies to Motion that repeats itself is very common and applies to rotation, to oscillating springs, to vibrating reeds, to waves rotation, to oscillating springs, to vibrating reeds, to waves as well as to many other phenomena. Whenever there is a as well as to many other phenomena. Whenever there is a net restoring force back to equilibrium for displacement, net restoring force back to equilibrium for displacement, we get oscillatory motion of one type or another. The we get oscillatory motion of one type or another. The oscillations can be simple (without change), with oscillations can be simple (without change), with diminished amplitudes (damping forces are present), diminished amplitudes (damping forces are present), increase amplitudes (resonance phenomenon in forced increase amplitudes (resonance phenomenon in forced oscillations), and variations such as coupled oscillations, oscillations), and variations such as coupled oscillations, etc.etc.


Ch 15: Oscillatory MotionCh 15: Oscillatory Motion

(15.1) Motion of Object Attached to a Spring(15.1) Motion of Object Attached to a Spring(15.2) Particle in Simple Harmonic Motion (15.2) Particle in Simple Harmonic Motion (15.3) Energy of Simple Harmonic Oscillator(15.3) Energy of Simple Harmonic Oscillator(15.4) Comparing Simple Harmonic Motion with (15.4) Comparing Simple Harmonic Motion with

Uniform Circular MotionUniform Circular Motion

(15.1) Motion of Object on Spring(15.1) Motion of Object on Spring

Resulting from force Resulting from force ∝∝ displacement from equilibrium.displacement from equilibrium.

If force acts toward the equilibrium position, a repetitive If force acts toward the equilibrium position, a repetitive backback--andand--forth motion results.forth motion results.

Examples:Examples:•• molecules in solidmolecules in solid•• e.m. waves e.m. waves •• ac circuit ac circuit •• pendulumpendulum

Forced oscillations replace lost energyForced oscillations replace lost energy

(15.1) Motion of Object on Spring(15.1) Motion of Object on SpringA particle moving in A particle moving in ‘‘xx’’ direction (horizontal springdirection (horizontal spring--

mass system).mass system).

(15.1) Motion of Object on Spring(15.1) Motion of Object on Spring

A particle moving in A particle moving in ‘‘xx’’ direction (horizontal direction (horizontal spring/mass system). Figure 15.1spring/mass system). Figure 15.1

( ) cos( )x t A tω δ= + A A ≡≡ max Amplitudemax Amplitudeωω ≡≡ angular frequencyangular frequencyδδ ≡≡ phase constant, phase constant,

(shift, angle)(shift, angle)t t ≡≡ timetimeT T ≡≡ periodperiod


(15.2) The Block(15.2) The Block--Spring SystemSpring System

HookeHooke’’s Law from chapter 7:s Law from chapter 7:

F k x= − ≡≡ linear restoring forcelinear restoring force

F k x ma= = ka xm

⇒ =

a x∝

displacementdisplacement

(15.2) The Block(15.2) The Block--Spring SystemSpring System2

2

d xdt

=k xm

= 2kletm

ω≡

2a xω=

dvadt

=


12T

π=2 fω π=

2 mk

π=1Tf

= 2πω

=12πω

=


Previous slide:Previous slide:

2 k km m

ω ω= ⇒ =

2

1 1 12

kf fT mπ ω π

= = = =

(15.3) Energy of Simple Harmonic (15.3) Energy of Simple Harmonic OscillatorOscillator

Without friction, energy is conserved, so:Without friction, energy is conserved, so:

2 21 12 2TE K U mv kx= + = +

At the maximum displacement, v = 0:At the maximum displacement, v = 0:

So,So, 212TE kA= MAXA x⇒ ≡ ∆

Problem [15Problem [15--27]27]

(15.4) Compare SHM & Uniform (15.4) Compare SHM & Uniform Circular MotionCircular Motion

Rotation of coil in Magnetic Field Rotation of coil in Magnetic Field and generation of AC sine wave.and generation of AC sine wave.

Rotating pin on disk makes Rotating pin on disk makes shadow on wall.shadow on wall.

(15.3) Energy of Simple Harmonic (15.3) Energy of Simple Harmonic OscillatorOscillator

Figure 13.9Figure 13.9

Problem Problem I.P. I.P. [13.21][13.21]

(15.4) Compare SHM & Uniform (15.4) Compare SHM & Uniform Circular MotionCircular Motion

Rotation of coil in Magnetic Field and generation of Rotation of coil in Magnetic Field and generation of AC sine wave.AC sine wave.

Rotating pin on disk makes shadow on wall.Rotating pin on disk makes shadow on wall.

LightLightSourceSource

WallWall

ShadowShadow

SHM SHM DemoDemo

SHM SHM DemoDemo


Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• DC Circuits DC Circuits •• Magnetic FieldMagnetic Field•• InductionInduction•• InductanceInductance•• ac circuitsac circuits•• Harmonic MotionHarmonic Motion


Mechanical waves are disturbances in a medium which Mechanical waves are disturbances in a medium which propagate through the medium. Examples are sound propagate through the medium. Examples are sound waves, earthquake waves, water waves, etc. Other types of waves, earthquake waves, water waves, etc. Other types of waves, such as electromagnetic waves , only need the waves, such as electromagnetic waves , only need the vacuum in which to propagate. The particles of these vacuum in which to propagate. The particles of these waves (the photons) propagate with the wave, but do not in waves (the photons) propagate with the wave, but do not in the mechanical wave. On the atomic level we find still the mechanical wave. On the atomic level we find still another type of wave, a probability wave associated with another type of wave, a probability wave associated with the motion of particles. the motion of particles.


Ch 16: Wave MotionCh 16: Wave Motion

(16.1) Propagation of a Disturbance(16.1) Propagation of a Disturbance(16.2) Sinusoidal, Traveling Waves (16.2) Sinusoidal, Traveling Waves (16.3) The Speed of Waves on a String(16.3) The Speed of Waves on a String(16.4) Reflection & Transmission (16.4) Reflection & Transmission (16.5) Rate of Energy Transfer (on string)(16.5) Rate of Energy Transfer (on string)

(16.1) Propagation of a Disturbance(16.1) Propagation of a Disturbance

Mechanical WavesMechanical Waves::•• Elastic MediumElastic Medium•• Energy SourceEnergy Source•• A physical mechanism by way of which adjacent A physical mechanism by way of which adjacent

portion of the medium can influence each other.portion of the medium can influence each other.

Electromagnetic WavesElectromagnetic Waves: (radio, light, etc.): (radio, light, etc.)•• Will travel through a vacuum (more in Ch 34)Will travel through a vacuum (more in Ch 34)


•• Transverse waveTransverse wave: Medium is displaced : Medium is displaced perpendicular perpendicular to direction of travel.to direction of travel.

•• Longitudinal waveLongitudinal wave: Medium is displaced parallel : Medium is displaced parallel to to direction of travel. direction of travel.


Negative Negative ≡≡ to the rightto the rightPositive Positive ≡≡ to the leftto the left

( , ) ( )y x t f x v t= ± Where Where ““ f f ”” is a functionis a function

For fixed value of For fixed value of ““xx””: The wave function represents : The wave function represents the ythe y--coordinate of a point as a function of time.coordinate of a point as a function of time.

For fixed value of For fixed value of ““tt””: The wave function defines the : The wave function defines the curve showing the shape of the wave at time curve showing the shape of the wave at time ““tt””..

(16.2) Sinusoidal Traveling Waves (16.2) Sinusoidal Traveling Waves

•• FrequencyFrequency ( f ): How often a wave occurs( f ): How often a wave occurs

•• PeriodPeriod (T): Time for (T): Time for one cycleone cycle of wave to occurof wave to occur

First time to see these:First time to see these:

•• WavelengthWavelength ( ( λλ ): Length of wave): Length of wave

•• WaveWave velocityvelocity ( v ): How fast wave moves( v ): How fast wave moves

(16.2) Sinusoidal Traveling Waves(16.2) Sinusoidal Traveling Waves

-2-1

01

2

DistanceDistance

Amplitude (A)Amplitude (A)Wavelength (Wavelength (λλ))

DisplacementDisplacement


-2-1

01

2

TimeTime

Amplitude (A)Amplitude (A) Period (T)Period (T)

DisplacementDisplacement

(16.2) Sinusoidal Traveling Waves(16.2) Sinusoidal Traveling WavesVarious relationshipsVarious relationships::

2k πλ

≡Wave number:Wave number:wavesmeter

⇒

rads

⇒

ms

⇒

2 fω π=Angular velocity:Angular velocity:

v fλ=Rate, freq., & Rate, freq., & wavelength:wavelength:

cycles

⇒1fT

= Hz≡Frequency vs. period:Frequency vs. period:


Wave equation as a function of position and time:Wave equation as a function of position and time:

2( , ) sin ( )y x t A x v tπλ

⎧ ⎫= −⎨ ⎬⎩ ⎭

Simplify on next slideSimplify on next slide

Displacement of Displacement of the mediumthe medium

(16.2) Sinusoidal Traveling Waves(16.2) Sinusoidal Traveling Waves2 ( )x v tπλ

⎧ ⎫−⎨ ⎬⎩ ⎭2 2x vtπ πλ λ

⎡ ⎤ −⎢ ⎥⎣ ⎦2k x vtπλ

−

2 vkx tπλ⎡ ⎤− ⎢ ⎥⎣ ⎦

[ ]2kx f tπ= − kx tω= −


So, the original expressionSo, the original expression……

2( , ) sin ( )y x t A x v tπλ

⎧ ⎫= −⎨ ⎬⎩ ⎭

Simplifies to Simplifies to ……

{ }( , ) siny x t A kx tω= −



Partial differentiation yieldsPartial differentiation yields……

Maxyv Aω=

2Maxy

a Aω=

(16.3) The Speed of Waves on a String(16.3) The Speed of Waves on a String

Tvµ

=

And laterAnd later……

.freq ∝ T T ≡≡ Tension in stringTension in stringµµ ≡≡ mass per length of stringmass per length of string

So, velocity of mechanical wave determined by So, velocity of mechanical wave determined by properties of the medium through which it travels.properties of the medium through which it travels.


Superposition:Superposition:

Constructive InterferenceConstructive Interference

Destructive InterferenceDestructive Interference

(16.4) Superposition & Interference(16.4) Superposition & Interference

Resultant is algebraic sumResultant is algebraic sum

Two waves can pass through each other without Two waves can pass through each other without being destroyed or altered.being destroyed or altered.

(16.4) Reflection & Transmission(16.4) Reflection & Transmission

(16.5) Rate of Energy Transfer(16.5) Rate of Energy Transfer

2 212P A vµω=

Rate of energy transmitted by sinusoidal waves on a Rate of energy transmitted by sinusoidal waves on a string:string:

So, Power is proportional to: So, Power is proportional to:

Now, check the units:Now, check the units:2

2kg rad mP mm s s⎧ ⎫= ⎨ ⎬⎩ ⎭

Ck for power Ck for power unitsunits


Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• CoulombCoulomb’’s Laws Law•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• DC Circuits DC Circuits •• MagneticsMagnetics•• ac circuitsac circuits•• Harmonic Motion Harmonic Motion •• Waves Waves


Audible Audible compressionalcompressional or longitudinal waves, to which the or longitudinal waves, to which the human ear is sensitive, range from 20 Hz to 20,000 Hz. human ear is sensitive, range from 20 Hz to 20,000 Hz. Waves below 20 Hz are called Waves below 20 Hz are called infrasonicinfrasonic and those above and those above 20,000 Hz are called 20,000 Hz are called ultrasonicultrasonic. .


Ch 17: Sound WavesCh 17: Sound Waves

(17.1) Speed of Sound Waves(17.1) Speed of Sound Waves(17.2) Periodic Sound Waves(17.2) Periodic Sound Waves(17.3) Intensity of Sound Waves(17.3) Intensity of Sound Waves(17.4) The Doppler Effect (17.4) The Doppler Effect

(17.1) Speed of Sound Waves(17.1) Speed of Sound WavesSound is a:Sound is a:•• Mechanical WaveMechanical Wave•• Longitudinal WaveLongitudinal Wave•• CompressionalCompressional WaveWave

(17.1) Speed of Sound Waves(17.1) Speed of Sound Waves

The Speed of Sound in Various Media: The Speed of Sound in Various Media:

Notice velocity of sound Notice velocity of sound in airin air is is affected by temperature:affected by temperature:

(17.1) Speed of Sound Waves (17.1) Speed of Sound Waves in Airin Air

.(331 ) 1

273

presenttemp

presenttemp

Tmvs

= +

velocity @ velocity @ In In ooCC0 0 ooCC

(17.2) Periodic Sound Waves(17.2) Periodic Sound Waves

As a wave propagates through a gas, the pressure As a wave propagates through a gas, the pressure of the gas varies harmonically (periodically)of the gas varies harmonically (periodically)

max( , ) sin ( )P x t P kx tω∆ = ∆ −


(17.3) Intensity of Sound Waves(17.3) Intensity of Sound Waves

PowerIntensity IArea

≡ = 2Wattm

⇒

(17.3) Intensity of Sound Waves(17.3) Intensity of Sound Waves

100

10log II

β⎛ ⎞

= ⎜ ⎟⎝ ⎠

120 21.00(10)

WIm

−≡

Threshold of Threshold of hearinghearing

deciBeldeciBel:: Measure of IntensityMeasure of Intensity

Threshold of hearing:Threshold of hearing: Softest sound we can hearSoftest sound we can hear

Problem [17Problem [17--31] 31]

(17.4) The Doppler Effect(17.4) The Doppler Effect

A A perceivedperceived frequency shift due to relative motion frequency shift due to relative motion between source of sound and observer.between source of sound and observer.

CarhornCarhorn--11Examples:Examples:CarhornCarhorn--22

Visual aids from the Web:Visual aids from the Web:DopplerDoppler--33


' OBS

SRC

v vf fv v

⎛ ⎞±= ⎜ ⎟

⎝ ⎠∓

•• f f ’’ ≡≡ frequency heard (frequency heard (perceivedperceived) by observer) by observer•• f f ≡≡ actual frequency emitted by sourceactual frequency emitted by source•• v v ≡≡ velocity of sound in this materialvelocity of sound in this material•• vv0BS0BS ≡≡ velocity of observervelocity of observer•• vvSRCSRC ≡≡ velocity of source of soundvelocity of source of sound•• Setup coordinate system with direction Setup coordinate system with direction fromfromobserver observer toto source as positivesource as positive Problem [17.45]Problem [17.45]


Shock WavesShock Waves:: When the velocity of Source When the velocity of Source exceedsexceedsthe velocity of the wave.the velocity of the wave.

Mach #:Mach #: Ratio of Source velocity to wave velocityRatio of Source velocity to wave velocity

# svmachv

≡ source Jetsound sound

v vv v

= =


sins

vv

θ =•• θθ ≡≡ apex halfapex half--angleangle•• v v ≡≡ velocity of sound in the mediumvelocity of sound in the medium•• vvss ≡≡ velocity of sourcevelocity of source

And,And,

# svmachv

≡Careful: v in Careful: v in denominatordenominator


(17.2) Periodic Sound Waves(17.2) Periodic Sound Waves

max( , ) cos( )S x t S kx tω= −•• S S maxmax ≡≡ Max displacementMax displacement•• k k ≡≡ WavenumberWavenumber (waves per meter) [from (waves per meter) [from chch 16]16]•• ωω ≡≡ Angular frequency (Angular frequency (rad/srad/s) )

(17.1) Speed of Sound Waves(17.1) Speed of Sound WavesSound:Sound:•• Mechanical WaveMechanical Wave•• Longitudinal WaveLongitudinal Wave•• CompressionalCompressional WaveWave

PB VV

∆= − ∆Bvρ

=

B B ≡≡ bulk modulusbulk modulusProblem [17.1] Problem [17.1]


Revised: 7/30/2013Revised: 7/30/2013

Well, youWell, you’’re all in a dead heat for the manager position re all in a dead heat for the manager position thus far. My final decision will be based upon the thus far. My final decision will be based upon the swimsuit swimsuit competitioncompetition..


•• MechanicsMechanics•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• DC Circuits DC Circuits •• MagneticsMagnetics•• ac circuitsac circuits•• Harmonic Motion Harmonic Motion •• Waves Waves •• Sound Sound


When two or more waves traveling in a medium encounter When two or more waves traveling in a medium encounter one another, what happens? The principle that applies, if one another, what happens? The principle that applies, if the distortions due to the waves are not too great, is the the distortions due to the waves are not too great, is the superposition principle (SP). SP says that the combined superposition principle (SP). SP says that the combined effects of two waves is the vector sum of the displacements effects of two waves is the vector sum of the displacements due to each individual wave. due to each individual wave. In this chapter, weIn this chapter, we are interested in the application of the are interested in the application of the SP to harmonic waves and particularly how the waves SP to harmonic waves and particularly how the waves combine to give us standing waves. This will lead us into a combine to give us standing waves. This will lead us into a discussion of modes of vibration. discussion of modes of vibration.


Ch 18: Superposition & Standing WavesCh 18: Superposition & Standing Waves

(18.3) Standing Waves in a String(18.3) Standing Waves in a String(18.4) Resonance (18.4) Resonance

Interference of wavesInterference of waves

(18.3) Standing Waves in a String (18.3) Standing Waves in a String fixed at both endsfixed at both ends

2nn TfL µ

=And, from chapter 16And, from chapter 16……

v∝n =1, 2, 3,n =1, 2, 3,……ffn n ≡≡ nnthth harmonicharmonicff1 1 ≡≡ fundamental freq. (1fundamental freq. (1stst harmonic)harmonic)T T ≡≡ TensionTensionµµ ≡≡ mass / lengthmass / lengthL L ≡≡ Length of stringLength of string

Hint on web for Hint on web for HmwkHmwk [18[18--46]46]Problem [18.21]Problem [18.21]

(18.4) Resonance(18.4) Resonance

•• The resonant frequency of a body is its natural The resonant frequency of a body is its natural frequency of vibration.frequency of vibration.

•• A A ““systemsystem”” may have many resonant frequencies.may have many resonant frequencies.

•• Demonstrate:Demonstrate:•• 2 tuning forks2 tuning forks•• ““BeatBeat”” frequency of two speakersfrequency of two speakers•• Forced vs. damped oscillations Forced vs. damped oscillations

Interference of WavesInterference of Waves

Points of Points of destructivedestructiveinterferenceinterference

Points of Points of constructiveconstructiveinterferenceinterference

Interference of WavesInterference of Waves

[Out of phase][Out of phase]DestructiveDestructive

[In phase][In phase]ConstructiveConstructive

Speaker DemonstrationSpeaker Demonstration

“Bird’s-Eye”View

Walk along this line and measure distance

between “dead”spots.

Audio generator

Outside Demonstration

YoungYoung’’s Doubles Double--Slit InterferenceSlit Interference

θθ

[[OppOpp] = d sin ] = d sin θθ

∆∆ λλ = 1= 1m = 1 m = 1

[[OppOpp] = ] = ∆∆ [PL][PL]

dd

θθdd

LL

yy

∆∆ [PL]=[PL]=∆∆ λλ = 0= 0m = 0m = 0

For For ConstructiveConstructive Interference:Interference:sind mθ λ= m = 0, m = 0, ±± 1, 2, 3 1, 2, 3 ……

YoungYoung’’s Doubles Double--Slit InterferenceSlit Interference

For For DestructiveDestructive Interference:Interference:

12sin ( )d mθ λ= +

where m = 0, where m = 0, ±± 1, 2, 3 1, 2, 3 ……

Look at Homework Problem [18Look at Homework Problem [18--?]?]

(18.1) Superposition and Interference (18.1) Superposition and Interference of Sinusoidal of Sinusoidal WavesWaves

Resultant of two Resultant of two travelingtraveling harmonic (periodic) waves:harmonic (periodic) waves:

2 cos sin2 2

y A kx tφ φω⎧ ⎫ ⎛ ⎞= − −⎨ ⎬ ⎜ ⎟⎩ ⎭ ⎝ ⎠

AmplitudeAmplitude

φφ / 2 / 2 ≡≡ phase anglephase angle


(18.2) (18.2) StandingStanding WavesWaves

( )( )02 sin cosy A kx tω=AmplitudeAmplitude

•• The amplitude of oscillation of a given particle The amplitude of oscillation of a given particle depends upon depends upon ““xx””(position). (position).

•• The Max. Amplitude occurs @ sin The Max. Amplitude occurs @ sin kxkx = +/= +/-- 11•• i. e., i. e., 3 5kx = , , , .

2 2 2etcπ π π

02Maxand A A= JavaJavaProblem [18.13] Problem [18.13]


Revised: 7/30/2013Revised: 7/30/2013


•• MechanicsMechanics•• The Electric FieldThe Electric Field•• Electric Potential (V)Electric Potential (V)•• DC Circuits DC Circuits •• MagneticsMagnetics•• ac circuitsac circuits•• Mechanical Waves Mechanical Waves •• SoundSound•• InterferenceInterference

Sound waves and water waves are mechanical waves Sound waves and water waves are mechanical waves which require a medium through which to travel. which require a medium through which to travel. Electromagnetic waves (e.g., radio and light), are similar Electromagnetic waves (e.g., radio and light), are similar to mechanical waves, but in many ways they are different. to mechanical waves, but in many ways they are different. E.m. waves propagate through a vacuum so they are not a E.m. waves propagate through a vacuum so they are not a disturbance in the medium. Therefore, they are not d

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Physics with Calculus IInwsccmoodledemo.com/thoward/214PowerPoint.pdf · 2015-07-23 · [25.16] It takes 385 nJ of energy to push the charges apart. I’m doing work to push them