Capacitors and RC Circuits
Introduction
A charging RC circuit consists of a battery of EMF ε, a resistor of resistance R, and a capacitor of
capacitance C. The potential difference across the resistor and capacitor can be derived as follows:
1
ε = ΔVc + ΔVr
2
3
ε = Q/C + IR
ε = Q/C + (dQ/dt)R
Kirchoff’s voltage law applied to this
specific circuit
Definition of capacitance and Ohm’s law
Definition of current
Everything in equation 3 is a constant except Q. The solution to this “differential equation” requires
mathematics beyond the scope of this course, so it must simply be asserted.
4
5
6
7
8
Q = Cε(1 – e-t/τ)
τ = “time constant” = RC
ΔVc = Q/C
ΔVc = ε(1 – e-t/τ)
ε = ε(1 – e-t/τ) + ΔVr
ΔVr = εe-t/τ
Solution to equation 4, a “differential
equation”
Definition of capacitance
Equation 4 in equation 5
Equation 6 in equation 1
Algebra
For a discharging capacitor consisting of only a capacitor and a resistor, a similar analysis can be done.
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10
11
12
13
14
15
16
0 = ΔVc + ΔVr
0 = Q/C + IR
I = -1/(RC)*Q
dQ/dt = -1/(RC)*Q
Q = Qoe-t/ τ
τ = RC
ΔVc = Q/C
ΔVc = (Q0/C)e-t/ τ
ΔVc = Voe-t/τ
Kirchoff’s voltage law
Definition of capacitance and Ohm’s law
Algebra
Definition of current
Solution to “differential equation”
Definition of capacitance
Equation 13 in equation 14
Definition of capacitance
The purpose of this lab is to test the equations for potential difference for charging and discharging RC
circuits.
Physics is fun!
Equipment You Procure
 stopwatch or clock
 AA battery
 digital camera
 your favorite Justin Bieber song (optional)
 an assistant (highly recommended)
Equipment from Kits
 1 AA battery holder
 2 digital multi-meters (DMM)
 1 F capacitor
 Resistors near 100 Ω
 alligator clips
 wires with clips (don’t use nichrome, which are the bare wires!)
Experimental Procedures
RC Time Constant while Charging
1) Measure the resistance of your resistor using a
DMM.
2) Calculate the theoretical time constant for an
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
RC circuit using your measured resistance and
the listed value for capacitance of 1.0 F ±
10%.
Assure that the capacitor has little or no
charge by connecting its two prongs with a
wire for about 10 seconds.
Connect the following items in a single loop
as shown: resistor, capacitor, and battery
holder. While doing so, connect the minus
end of a battery holder to the minus end of
the capacitor which has a stripe on its side.
Configure two DMMs as voltmeters and place
them in the alligator clips so that they measure
the potential differences across the resistor and the capacitor.
Play your favorite Justin Bieber song. At this point, you may want to include an assistant.
Put the batteries in the holders. Start your stopwatch as you put the last battery in the holder.
Measure the potential differences across the resistor and capacitor every 10 seconds or so for
about 3 minutes.
Disconnect one of the wires or remove a battery.
Graph the potential difference as a function of time for the capacitor. Test if the graph looks
qualitatively like theory predicts. If so, then this is evidence in favor of equation 6.
Graph the potential difference as a function of time for the resistor.
For the graph of the resistor, add an exponential (not linear) trend line to obtain a curve and an
equation for potential difference. Increase the number of digits displayed in the equation
so that at least two significant digits appear in the exponent. Does this curve go through
most of the error bars? If so, then this is evidence in favor of equation 8.
Take the magnitude of the constant in the exponent of the equation from the graph and raise it
to the power of -1 to obtain an experimental value for the time constant. You will not be able
to obtain an error in this value. Compare this value to the theoretical value. If they overlap,
then this is evidence in support of equation 8.
RC Time Constant while Discharging
14) Repeat steps 7 through 12 when you connect the charged
capacitor in a loop with the resistor (no batteries) as shown.
You will only need one voltmeter in this case as the resistor
and capacitor will have the same potential difference. Begin
measuring immediately after connecting this circuit as the
capacitor will begin to discharge through the resistor.
15) If the trend line goes through the error bars, then this is
evidence in favor of equation 16.
16) If experimental and theoretical time constants match, then
this is evidence in favor of equation 16.
17)
Remember to increase the number of
digits displayed in the equation for the
graph so that at least two significant digits appear in the
exponent.