University of Puget Sound Introductory Physics Laboratory
7. Time-dependent circuits
Name:____________________
Date:___________________
Objectives
To investigate the time-dependent behavior of circuits involving capacitors and
resistors.
Equipment
Capacitors, resistance boxes, voltmeter, light bulbs, and semi-log graph paper.
Helpful Reading
Review the introduction to potential difference, current, and resistance in either
Hecht Chapter 17 an 18 or Rex/Jackson Chapter 21.
Background
In the last lab session you analyzed circuits involving resistors using the node
and loop equations. For circuits with other kinds of elements (capacitors,
inductors, transistors, etc.) we can apply the same analysis provided we know
how to describe the voltage across the various circuit elements. For resistors, the
potential difference across the resistor is given by Ohm’s Law V=IR, where R is
the resistance and I is the current through the resistor. Today we experiment
with capacitors, which are simply parallel metal plates separated from each other
by some non-conducting material. The circuit symbol for a capacitor is two
parallel lines with a small separation between them (this is to remind you of
parallel plates).
+ polar
non-polar
When current flows into a capacitor, it can't jump to the other plate so charge
piles up on the plate. This charge creates an electric field between the plates, and
hence a voltage difference across the capacitor. The potential difference across a
capacitor is proportional to the charge on the plates
V = Q/C
7-1
where Q is the charge on the capacitor (+Q on one plate, -Q on the other plate)
and C is the capacitance. Capacitance is measured in Farads, which is another
name for a Coulomb per volt.
It is important to note that there are two kinds of capacitors, polar and non-polar. The
non-polar capacitors, shown above on the left, can be used in any orientation. Their
terminals are equivalent. Polar capacitors, on the other hand, have a polarity (+, -)
associated with the terminals and must be hooked up correctly in order to work (and not
explode). We will be using polar capacitors. Make sure that the negative (low potential)
terminal of your battery or power supply will be connected to the negative terminal of the
capacitor (an arrow with a negative sign indicates the capacitor’s low potential terminal).
Charging and discharging experiments with light bulbs
Assemble the charging circuit shown below with two batteries (but don't connect
the batteries yet), a light bulb, and a 4700 F polar capacitor. Now connect the
batteries and watch the bulb. What happens?
light bulb
3 volts
(2 batteries)
4700F
+
-
Charging
circuit
4700F +
Discharging
circuit
Description/discussion of charging behavior
7-2
The capacitor should now be charged to 3 V - right? Disconnect the batteries and
use the charged capacitor to make a discharging circuit with the light bulb. What
do you observe? Does this differ from what you observed while charging?
Repeat the charging/discharging process a few times; see if you can fine-tune
your observations.
Description/discussion of discharging behavior
Think about what you have observed in terms of the movement of charge in the
circuit. Assume that you have a capacitor C initially charged to a potential
difference V0. Now connect it across a resistor R. Why does the current flow?
Why does the current stop?
C
R
I
Capacitors store energy. Starting with a charged capacitor, identify the forms
that the energy takes from fully charged state to the fully discharged state.
Discussion of charge movement and forms of energy
7-3
Derivation of the potential difference across a discharging capacitor
We can formalize your observations mathematically to model the discharging
circuit behavior. The rate at which charge is removed from the capacitor must
equal the current that flows through the resistor:
I = -dQ/dt
From the loop equation we know that the voltage difference across the capacitor
and the resistor must be equal. Using V=IR for the resistor and V=Q/C for the
capacitor, the above equation becomes:
dV
1

V
dt
RC
This is a differential equation for the potential difference across the capacitor as
function of time. We can solve this by rewriting the equation as
dV
1

dt ,
V
RC
which can be integrated from 0 (initial time) to t (later time) on the right side and
from Vo (initial potential difference) to V (potential difference at the later time) on
the left to yield
V(t) = V0 e-t/RC
The voltage (and hence the charge) on the capacitor decreases exponentially in
time, with a characteristic time  = RC. This is amazing: R is in Ohms and C is in
Farads. In the space below, prove that an Ohm-Farad is indeed equal to a
second. By what factor is the voltage reduced after the first  seconds? After the
second  seconds?
dimensional analysis
7-4
Discharging experiment
By charging a capacitor and measuring the potential across the capacitor while it
discharges, we can quantify the discharging behavior and compare it to the
exponential decay model. Construct the following circuit, taking special care about
the polarity of the capacitor. Exploding a 4700 F capacitor would be about as
memorable as the earthquake. Be sure the negative terminal of the capacitor is
connected to the negative terminal of the power supply.
B
r
+
power
supply
-
D
A
C
+
-
V
R
The resistor R and capacitor C comprising our RC circuit are on the right. The
knife switch ABD is used to charge and then discharge the capacitor. The power
supply is used to charge up the capacitor initially by putting the switch in the
“AB” position. (The resistor, r =10 , is there to prevent the capacitor from being
ruined by a large current from the supply.) After the capacitor is fully charged,
the switch is moved to the “AD” position and the voltmeter is used to measure
the potential difference across the capacitor as a function of time.



Use a variable resistance box set at 4000  for R and one of the 4700 F
capacitors for C. What decay time would you predict?
Put the switch to “AB” and adjust the power supply to charge the capacitor
up to 5 V.
Move the switch to “AD” and measure the voltage across the capacitor as a
function of time.
Do a few practice runs to get a sense of how long a run you will need and how
often you will be able to take data. Work out a plan with your group members
as to who will do what and when. Blank data tables are provided on a following
page. You may need to do this more than once to get a good data set. Make sure
to take data over several 'e-foldings', that is, over a time longer than several RC
times.
Plot your data on semi-log paper, not on the computer, with voltage on the
logarithmic axis and time on the linear axis. Why are we plotting with log-linear
7-5
axes? Take the natural logarithm of the V(t) expression given above. If you were
to plot ln(V) vs. time, what would it look like? Does your data look like this?

Obtain voltage vs. time data sets for the following resistor /capacitor
combinations:
4000  / 4700 F (just completed)
2000  / 4700 F
4000  / 9400 F
You do not have a 9400 F capacitor. How can you make one with two that are
4700 F? Plot these results on the same semi-log paper that contains your other
graph.
What are the physical significance of the slope and intercept? Do the measured
values of slope and intercept agree with your experiment? (You will need to be
careful when obtaining slope and intercept information from your plot.)
Check-in with your instructor as you leave:
1. Voltage/time data.
2. Completed voltage/time graphs.
3. Explain the concept of a decay time, and how you read one from a semi-log
graph.
7-6
Time (sec)
Voltage (volts)
Time (sec)
7-7
Voltage (volts)