Physics 261
Simple Resistor-Capacitor Circuits
Having investigated both capacitors and resistors, as well as simple DC resistor circuits, you will
now look at circuits with both resistors and capacitors. In particular, you will test the equation that
describes the discharge through a resistor of a charged capacitor.
The potential difference between the electrodes of a capacitor with capacitance C storing a charge
Q is given by
Q
V = .
(1)
C
When the capacitor discharges through a resistor with resistance R the rate that it does so must
equal the current through the resistor:
dQ
= −I
(2)
dt
The minus sign indicates that when the charge is being reduced, current leaves the capacitor.
Using equations 1 and 2, and Ohm’s law, you should be able to show (do so in your notebook)
that
dV
V
=−
(3)
dt
RC
This is a first-order, linear differential equation. Carry out the steps in your notebook to solve this,
and get
t
V (t) = V0 e− RC ,
(4)
assuming V (t = 0) = V0 .
Equation 4, the so-called RC decay equation, is the equation you will be testing. Note that time
is the independent variable and voltage is the dependent variable.
Referring to equations 1 and 2 and Ohm’s Law, deduce the units of the product RC, which is
known as the time or decay constant of the circuit.
Slow RC Decay
Choose a 10 kΩ resistor and a 1 µF capacitor, and check that they are within specifications.
When you have a pair that are, assemble the following circuit, that also includes a power supply
and a voltmeter.
Note that the voltmeter is in series with the circuit, as it’s impedence (high resistance) will be part
of the the total resistance of the circuit. You should measure the resistance of the voltmeter with
an(other) ohmmeter.
Calculate the total resistance: RT = Rresistor + Rvoltmeter and propagate uncertainties.
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Calculate RT C and propagate uncertainties.
Turn on the power supply to roughly 10 volts. You don’t need a precise measurement or uncertainty.
You will disconnect the power supply by removing its cables from the circuit–not by turning it
off and read off the voltmeter at regular intervals, of say 2 seconds. This has to be done in teams
of three, with one person looking at a stopwatch and calling out when to read the voltage, one
person reading the voltmeter out loud, and one person recording the values. Voltage and time
values comprise your data.
Analyze your data set in Excel, determining the form of the relationship and the value of the
exponential term. The form of the relationship should correspond with that of equation 4, and there
should be a way to compare the exponential term to the decay constant you calculated. All these
must agree satisfactorily for the equation to be verified.
Fast RC Decay
Let V (t) = V0 /2 and solve (in your notebook) equation 4 for t. This special value of t = t1/2 is
known as the half-life of the circuit.
You will measure the half-life of an RC circuit directly and compare it with a calculated value.
Using your 10 kΩ resistor and 1 µF capacitor (but not the power supply or voltmeter), set up the
following circuit, which includes a wave generator and an oscilloscope (note, the plus and minus
cables go into the same oscilloscope input channel).
The wave generator should be set to output a square wave with Vp−p = 5 V. When the wave is
positive, the capacitor will charge, and when it is at ground, the capacitor will discharge. The
period of the square wave therefore must be long enough to allow full charge and discharge, each
about 4 time constants. Recalling the relationship between period and frequency, set the frequency
of the wave generator appropriately. You should see on the oscilloscope signals like those in the
figure below.
Determine the time it takes for the a signal to decrease from its maximum to half its maximum.
This should be the same as the half-life. Check.
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