RC Circuit Exponential Decay
Purpose: to demonstrate the exponential decay of the voltage across an RC circuit; to discover the relation of the time constant of the decay to nuclear physics concepts such as halflife and lifetime
Materials: 9 volt battey, 9 v battery clip, resistor (~1 megohm), capacitor (~10 mF), connecting wires, timer
Procedure:
1. Connect the capacitor and resistor parallel to each other; then connect the battery and clip to the capacitorresistor combination. See the schematic diagram and the sketch in Figure 1. If the capacitor is polarized, make sure that the black (+) lead of the clip goes to the + side of the capacitor.
2. Wait for 25 minutes. During this time the capacitor will charge up.
3, Set the multimeter to read voltage.

Initial Circuit 
4. Disconnect the battery from the clip and immediately connect the multimeter to the clip. See Figure 2. Start your timer and record the voltage for time 0 in your data table. A data table is given in the student worksheet.
5. Every ten seconds, read and record the voltage. Continue this for at least two minutes or until your voltage readings change only very slowly from interval to interval.
6. Make a graph of voltage as a function of time. (See the student worksheet.)

Final Circuit 
Analysis:
Determine if the curve you get in your graph truly indicates exponential decay. One way to do this is to graph the natural logarithm of the voltage as a function of time. First, divide every voltage V by your initial voltage V_{o}. Then find the natural logarithm of V/V_{o} for each time. Finally, graph ln (V/V_{o}) as a function of time. Do you get a straight line with a negative slope? If you do, then you have an exponential decay according to the equation V = V_{o} e^{t/t}, where t is the time constant of the RC circuit.
The decay of elementary particles and the decay of nuclear isotopes follow the same mathematical form. In particle physics, the particle lifetimetakes the place of the time constant. In nuclear physics, the halflife is the time for the number of atoms of the isotope to decay to half its value; how would you find the halflife of our RC circuit?

Student Worksheet
Data Table:
Time t (sec)

Voltage V (volts)

V/V_{o}

ln (V/V_{o})

0
10
20
30
40
50
60
70
80
90
100
110
120


1

0

Make an additional or a longer table if data runs beyond 120 sec.
Graphs:
