RC Circuits 0

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Department of Physics & AstronomyUndergraduate Labs

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1/2/2012

RC Circuits

When a capacitor is connected to a voltage source, charge will

accumulate on the plates of the capacitor until the voltage

across the capacitor is equal to the voltage across the source.

The

charge

on

the

capacitor

plates

is

proportional

to

the

voltage across the capacitor’s terminals. The constant of

proportionality is the capacitance .

Capacitance is a function of the geometry of the capacitor and the

material properties of the dielectric that is sandwiched between the

plates. For a capacitor with plate area , separation containing a

dielectric between the plates of dielectric constant , the capacitance is

Real capacitors (figure on the right) oftentimes look like plastic cans as

opposed to parallel plates. However, the plastic is simply insulation and the

inside consists of long strips of conducting foil wound around a dielectric. This

configuration enables a large surface area of conductor to be stored in a

small amount of space.

Capacitors are useful for many reasons, but most of all because they can be used to store electrical

energy in the form of a static charge

on the plates. Capacitors can be charged by connecting them to a

voltage source and discharged at a later time. However, capacitors cannot be charged or discharged

instantaneously. The amount of time to charge/discharge depends on the resistance of the circuit.

Below is a diagram of a capacitor being charged by a voltage source. During charging, current will flow

through the resistor as negative charge accumulates on the bottom plate, which repels an equal amount

of charge off the top plate. Charge will accumulate until the voltage across the capacitor is .

/




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Summing voltage drops around the circuit gives

0

This can be rewritten (substituting /) as the differential equation

0

This has the solution

1 / Recognizing that is the maximum charge that can be stored on the capacitor we can write

1

The quantity is the RC‐time constant that dictates how fast the capacitor can be charged. The

figure below shows a few plots of / for various values of the time constant .

Disconnecting the capacitor from the voltage source leaves the capacitor with a charge on the

plates. This charge can be used at a later time to power some electric device, for example. When the

capacitor is discharged, as shown below, a current will flow through the resistor as electrons flow from

the – plate to the + plate. This will continue until the charge on both plates is zero.




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Similar to before, we can sum voltage drops around the circuit:

0

This can be rewritten (substituting /) as the differential equation

This has the solution

/

Again, we see the quantity dictates how fast charge on the capacitor decays. The figure below shows

a few plots of / for various values of the time constant .

In this lab you will observe charge flowing to and from the capacitor in real‐time and observe similar

plots to those shown above.

Goals of this Lab

1. Understand how a capacitor obtains and stores charge

2. Understand how charge flows through a circuit

3. Understand how the capacitance and resistance of a circuit affects charge flow

4. Observe charge flow in real‐time




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Lab Materials

Power Supply

The power supply will act as a battery that can provide a variable

(i.e. user defined) voltage.

Banana Connector Patch Cord

Electrical connections can be made by the sticking the plug of one

into the socket of another.

Alligator Clips

Alligator clips will be used to make electrical contact in your circuit.

Switch

Electrical

connection

to

two

different

circuits

will

be

made

via

a

switch.

Multimeter

The multimeter will primarily be used for troubleshooting your

circuit and measuring resistances.

Differential Voltage Probe

These can be interfaced to LoggerPro for automatic data collection

of voltage differences in a circuit.

LoggerPro

LoggerPro will be used to run the differential voltage probe and used

for data analysis. LoggerPro

Resistors

Resistors will be used to regulate the rate of charging and

discharging a capacitor.

Capacitors

Charge from the power supply will be stored in capacitors.

Light bulbs

Light bulbs will be used to ensure that circuits are properly

connected.




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Overview

You will build the circuit shown below. The circuit consists of two branches and a central capacitor .

The left branch contains a voltage source and a resistor . The right branch contains a resistor

only. The capacitor can be connected to either branch by throwing a switch to the left or right. Closing

the switch

on

the

left

branch

connects

the

capacitor

to

a voltage

source

and

will

charge

the

capacitor.

After the capacitor is charged, the switch can be opened. Charge is now stored on the capacitor. The

capacitor can be discharged by closing the switch on the right branch.

During both charging and discharging you will monitor the charge on the plates by measuring the

voltage across the capacitor. You will connect a voltage probe across the terminals of the capacitor as

shown below

The voltage is related to charge via the relationships . Thus, the voltage will also show an

exponential rise or decay governed by the time constant as the capacitor is charged or discharged,

respectively. We will exploit this to measure the capacitance of several different capacitors by observing

these exponential rises and decays of voltage.

Voltage

To computer




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Lab Procedure

Set up the power supply and LoggerPro

The voltage probes max out at 6V so we want to make sure we apply a lower voltage than this. If you

have a Vernier power supply (small black box), it automatically outputs 5V so go to step 4. If you are

using the PASCO Power Supply follow steps 1‐3:

1. If you are using the PASCO Power Supply set the function generator to DC mode by clicking the

downward arrow button () on Range until the display shows 00000.

2. Set the multimeter to DC voltmeter mode and connect probes to the terminals of the power

supply.

3. Dial up the voltage on the power supply to 5.0V. Leave the dial at this setting!

4. Run LoggerPro and set up the experiment for continuous data collection at 100 samples/s. Go to

Experiment

Data

Collection…

and

check

Continuous

Data

Collection

and

set

the

Sampling

Rate to 100 samples/s.

Build a branched circuit that can charge or discharge a capacitor

1. Choose two resistors and and a capacitor and build the circuit shown below. Measure

the values of and with the multimeter.

Note: You will use only one switch. The diagram shows two to help you visualize how the

physical switches work. When the physical switch is closed, it is equivalent to closing the top and

bottom switches in the circuit diagram below.

Note: Some of the capacitors we are using are electrolytic capacitors that use an electrolyte

(instead of metal) for one plate. These capacitors have a polarity, meaning that one side must be

connected to the positive terminal of the power supply. This terminal is marked by a +.

2. Connect the voltage probe as shown and observe the voltage displayed in LoggerPro. The

voltage across the capacitor should be zero (with small fluctuations about zero). Discharge the

capacitor by shorting out the terminals with wire if necessary.

Voltage

To computer




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Prelab Exercise 1:

You are given the following set of capacitors and resistors. Which combination will cause the capacitor

to charge at the fastest rate when you close the left‐hand switch? What resistance should you use if

you want a capacitor to charge at the slowest rate? What resistor should you use if you want a

capacitor to discharge at the fastest rate?

Capacitance

Resistance

130×106 µF 500

150 µF 1 M

1500 pF 10 k

Verify that your circuit works as expected

1. Replace and with light bulbs.

2. Throw the switch to the left. You should observe a flash of light through the bulb as your

capacitor is charged. Depending on how large

is, the flash may last for several seconds.

3. Close the switch to the right. You should observe a flash of light with identical (provided that the

light bulbs are about the same) duration.

4. If you observed the flashes, your circuit is properly connected. Replace the light bulbs with

and .

Prelab Exercise 2:

Why should you observe only a flash of light, as opposed to a continuous glow, when the switch is

thrown to the left? After all, the light bulb is directly connected to the power supply.

Charge the

capacitor

1. Start collecting data with .

2. Close the switch to the left to charge the capacitor. Observe the voltage vs. time. Autoscale the

graph with as needed.

Note: Depending on your values of and , the capacitor could charge very fast or very slow.

If nothing happens, something in your circuit isn’t connected properly.

3. Wait until your capacitor is fully charged (i.e. until the voltage is constant). This typically takes

about 5 time constants (5. Don’t stop collecting data!

Prelab Exercise 3:

How many time constants (give an integer number) will you have to wait before the voltage across the

capacitor is within 99% of its initial value?




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Discharge the capacitor

1. Throw the switch to the right to discharge the capacitor. Observe the voltage vs. time.

Note: Again, depending on your values of and , the capacitor could discharge very fast or

very slow. If nothing happens, something in your circuit isn’t connected properly.

2. Wait until your capacitor is fully discharged (i.e. until the voltage is about zero). Again, this takes

about 5 time constants (5). Stop data collection ( ) when it is fully discharged.

Analyze your results

You should observe two curves showing the charging and discharging of the capacitor. They should look

similar to the curves in the introduction.

1. Fit a Natural Exponential ( ) to the voltage rise curve (charging). Use the brackets [ ] on

either side of the data to select the region for the fit. You may have to rescale the horizontal axis

to zoom

in

on

the

curve.

Do

this

by

left

clicking

the

first

and

last

tick

label

of

the

horizontal

axis

and changing the value. Don’t include the flat region to the left of the curve in the fit!

2. Fit a Natural Exponential ( ) to the voltage decay curve (discharging). Use the brackets [ ] on

either side of the data to select the region for the fit.

3. Determine the capacitance of your capacitor from each of your exponential fits.

4. Create a column that calculates the charge by going to Data New Calculated

Column and entering in the appropriate values in the Equation box. Use the average value

you obtained from the previous step, not what is marked on the side of the capacitor! You can

use

values

from

columns

that

contain

data

(i.e.

the

potential

column)

by

clicking

.

5. Repeat this experiment using a different combination of and values. If you didn’t do so,

choose values that will fully charge/discharge the capacitor over the timescale of minutes.

Questions

1. Attach plots of the charging/discharging curves including fits. Label the resistances for each curve.

2. Make a table that lists the resistances and and your measured value of capacitance . 3. Do your capacitance values match for charging and discharging? If not, what could be causing the

discrepancy? Do your capacitance values match with what is labeled on the side of the capacitor?

4. What was the maximum charge

stored on each capacitor? How many electrons does this

correspond to?

The

charge

of

an

electron

is

1.6 10C.

Feedback

Please leave feedback about this lab. What did you like best? What did you like least? Did anything go

wrong? What did you feel like you learned in this lab?

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