INTRODUCTION

    The purpose of this experiment is to gather data by charging and discharging capacitors in a series RC circuit, then to examine some of the properties of exponential functions.

THEORY

Exponential functions occur in all branches of science; here are some of their important properties:

1. They have a characteristic time often denoted by tau (τ).

2. In any time interval equal to tau, a decaying exponential decreases to 37% of its initial value.

3. An increasing exponential rises from zero to 63% of its final value in time tau.

EXPERIMENT

1. Connect the circuit shown so that you can charge and discharge the capacitor.

2. Charging: Set your voltmeter so that it reads to 0.01 volts DC. Measure the voltage across the capacitor every 10 seconds for 200 seconds after closing the switch to charge the capacitor. Leave the voltage on and record its value after it stops changing (V_max). Record your data using a table as shown.

3. Discharging: Repeat step 2, moving the switch so that you discharge the capacitor, recording the voltage across the capacitor every 10 seconds.

ANALYSIS

4. Plot three graphs of your data (all as a function of time): the charging voltage across the capacitor; the discharging voltage; and the ln (natural log) of the discharging voltage.

5. Find values of tau from (i) your charging graph (the amount of time to 63% of V_max), (ii) the discharging graph (time to 37% of V_max) and (iii) from the slope of the log graph. For (iii) you need to know that
   1. V_c = V_maxe^{( –t/_tau )}

   2. lnV_c = lnV_max – t( ¹/_tau )

   Compare step (b) with y = b + mx (for a straight line), and calculate tau from the slope.

6. Find the expected value of tau from τ = RC (Note: you are using 500 microFarad capacitors. Some are marked 500 µF, while others use the older notation of 500 mFd. Both are equivalent!).

7. Use the computer to determine a value for tau, using both the charging and discharging data. Since it has been awhile since the computer was used, your instructor will assist you.

REPORT

• Present your six values for tau; you should clearly indicate the source of each result.

• Discuss if the six estimates for tau are consistent; if not, try to explain why.

• What assumptions did you have to make in step 6? How can you check your assumptions? Be sure to check.

• Finally, use the log graph and the fit of the computer line to your data to show whether or not your data for the discharge curve are exponential (explain your logic!).

©	St. Lawrence University	Department of Physics

Revised: 16 May 17

Canton, NY 13617