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RC CircuitsRC CircuitsFrom: PASCO Exp. 012-05892A

Objectives: To study how the voltage across a capacitor varies as it charges and discharges, and to findthe capacitive time constant of an RC circuit.

Apparatus:Computer with Science Workshop™InterfacePower amplifier (CI-6552-A)Voltage Sensor (CI-6503)RLC Circuit Board (CI-6512)(2) Banana plug patch cords

Background:

When an uncharged capacitor isconnected across a DC voltage source, the rate atwhich it charges up decreases as time passes. Atfirst, the capacitor is easy to charge because thereis very little charge on the plates. But as charge accumulates on the plates, the voltage source must “domore work” to move additional charges onto the plates because the plates already have charge of thesame sign on them. As a result, the capacitor charges exponentially, quickly at the beginning and moreslowly as the capacitor becomes fully charged. The charge on the plates at any time is given by:

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q t( ) = qo 1− e− tτ

where qo is the maximum charge on the plates and τ is the capacitive time constant (τ=RC, where R isthe resistance and C is the capacitance.)

NOTE: The stated value of a capacitor may vary by as much as ±20% from the actualvalue. Taking the extreme limits, notice that when t=0, q=0 which means there is not anycharge on the plates initially. Also, notice that when t goes to infinity, q goes to qo whichmeans it takes an infinite amount of time to completely charge the capacitor.

The time it takes to charge the capacitor to half full is called the half-life and is related to thetime constant in the following way:

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t 12

= τ ln2

In this experiment the charge on the capacitor will be measured indirectly by measuring thevoltage across the capacitor since these two values are proportional to each other: q=CV.

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Procedure:

PART I: Computer Setup

1) Connect the Science Workshop™ interface to the computer, turn on the interface, and turn onthe computer.

2) Open DataStudio and create a new experiment.

3) Click the SETUP button, , at the top and then pick a voltage sensor from theSENSORS categories and drag it to the “Analog A” icon on the picture of the interface.Double click on the sensor icon and set the sensor properties to “low sensitivity” and 4000Hz sampling rate.

4) Below the sensors list is a signal output. Drag this icon to the “output” ports on the far rightof the interface. Double click on the output icon. Set the output to a positive square wave of4.000 V amplitude and 1.000 Hz frequency.

5) Drag a graph icon from the DISPLAYS window onto the “voltage” in the DATA window.

PART II: Equipment Setup

1) Construct the circuit shown to theright on the RLC Circuits Board.This is done by, placing a jumperwire between the 100 Ω resistor andthe 330 µF capacitor. Also connectone end of the RC pair to the (+)terminal of the power supply andthe other end to the (-) terminal.

2) Connect the Voltage Probe to theconnectors at both ends of the 330µF capacitor.

PART III: Data Recording

1) Click start, . After 1-2 seconds, with 1-2 cycles displayed on the screen stop thedata collection.

PART IV: Analyzing the Data

To power (+) To power (-)

R to C connection

To Voltage Probe

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1) Click the “Autoscale” button ( ) in the Graph to rescale the Graph to fit the data.

2) Click the “Magnifier” button ( ). Use the cursor to click-and-draw a rectangle over aregion of the plot of Voltage versus Time that shows either the voltage rising from zero to themaximum volts, or from maximum voltage back to zero.

This will give you an expanded view of the Voltage versus Time for that region.

3) Click the “Smart Cursor” button ( ). A cross-hair should appear on the graph.

4) Move the cursor to the point where the voltage begins to rise (or fall). Record the time that isshown.

5) Move the “Smart Cursor” to the point where the voltage is approximately one half themaximum number of volts. Record the new time that is shown.

6) Find the difference between the two times and record it as the time to half the maximumvoltage, or t1/2.

7) Having a known value of the half-life and an accurate value for the resistance, use theequation from the beginning of the lab to determine the actual capacitance of the capacitor,C.

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8) Repeat the procedure for other values of R and find an average value for the actualcapacitance of the “330 µF” capacitor.

9) If you have time, find the actual value of the capacitor labelled “100 µF” on the RLC circuitboard.

QUESTIONS: (Answer these questions in reference to your first circuit: R=100 Ω, C=330 µF)

1) The time to half-maximum voltage is how long it takes the capacitor to charge halfway.Based on your experimental results, how long does it take for the capacitor to charge to 75%of its maximum value?

2) After four “half-lifes,” to what percentage of the maximum charge is the capacitor charged?

3) What is the maximum charge for the capacitor in this experiment?

4) What are some factors that could account for the percent difference between the stated andexperimental values?