RC Circuits
6. RC Circuits*
Objective: You will become familiar with the fact that the current flowing to or from a capacitor
plate is the same as charge flow onto or off from the plate. You will see that in simple RC circuits
the voltages and currents vary exponentially in time. As an application your team is asked to
solve the practical problem of measuring the resistance of a multi-meter.
The learning objectives are the following:
1. To be able to explain how charge on either plate of a capacitor relates to the
current flowing to or from the plate.
2. To understand why the internal resistance of a meter must be taken into account
when measuring voltage across a circuit element with large resistance.
3. To recognize the time scale for changes in voltage and current in a circuit
comprising a resistor and capacitor; to see that it is controlled by the product RC,
which is a time constant.
4. To apply this understanding to a practical problems in electrical measurement.
(These concepts are described in your textbook.)
Reading assignment:
Review capacitors, circuits and RC circuit. Read the following sections. (Section numbers may be
slightly different depending on the edition of your textbook: Check the section titles.)
Knight, Jones and Field : 21.7 Capacitance and Capacitors, 23.7 RC circuits.
Serway and Vuille (212): 16.6 Capacitance, 16.7 Parallel Plate Capacitor, 18.5 RC Circuits
Serway and Jewett (252): 26.1 Definition of capacitance, 28.4 RC Circuits
Pre-lab exercises: Bring the answers to the following questions with you to lab. The instructor
will check them when you enter. You will turn them in finally with your report.
1. Before S is closed, the capacitor C is uncharged. When it is closed, a charge begins to build
up on the upper plate.
(a) What is the sign of the charge building up on the upper plate?
b
S
a
(b) Which way do the electrons move in
the upper wire?
B
c
C
(c) Does a charge accumulate on the lower plate
as well? If so what sign does it have?
R
d
(d) As the charge builds up on the lower plate,
which way do the electrons move in the lower
wire?
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*© William A Schwalm 2012
RC Circuits
(e) In both the top and the bottom wires the current flow is in the direction opposite to that of the
electron motion. Why is that?
(f) To charge the capacitor, current flows out of the positive terminal of the battery (top). But an
equal current must flow into the negative terminal. Since no charge can flow across empty space
between the two capacitor plates (disks in the figure) how does charge get from the positive
battery terminal all the way around to the negative terminal? Explain.
2. Consider again the same circuit as in problem 1.
(a) After the switch S has been closed, and the capacitor is partly charged (not fully charged) how
is the voltage difference Va – Vd across the capacitor related to the charge Q on the upper plate of
the capacitor? Write an equation.
(b) At the same instant as in part a), how is the voltage difference Vd – Vc across the resistor
related to the current I flowing through the resistor? Give an equation.
(c) How is the voltage difference Vb – Vc related to the battery voltage Vo ?
(d) Use your responses to the parts above to come up with an equation involving Q and I and the
battery voltage Vo.
(e) Looking at the equation you have written, explain why it is that, as the charge Q builds up on
the capacitor plate, the rate of charging decreases. (What variable represents the rate of
charging?)
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RC Circuits
(f) If you were to wait a very long time after closing S, what would be the limiting values of the
charge Q on the upper capacitor plate, the voltage Vd at the other plate, and the current I ?
3. (Part c is for Phys 212 and 252) Suppose a multi-meter is used to measure voltage differences
in the circuit below, represented as a schematic diagram. Suppose the battery is 10 Volts and
each resistor is 47 mega Ohms.
(a) Without applying the meter, what should be the voltage difference between points b and c?
(b) If you want to measure the voltage difference from point b to point a (a positive value) where
would you connect the V  wire and where would you connect the COM wire? Indicate these
connections with lines.
R1
a
(+)
b
OFF
V
V
c
A
R2

V 
A
COM
A
10A
(c) Now suppose you actually make this measurement and find
Va  Vb  4.89 Volts .
What is the resistance of the volt meter? (Actually, the effective volt-meter resistance will depend
on which scale you have it set to, so this would be for a particular meter setting.)
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RC Circuits
Scenario: The development team at Ryan Measurement Technologies is assigned to write
instructions for using standard resistors and a multi-meter to measure capacitance. As usual they
have passed this responsibility to your group. The idea is simple, but to prepare for the job, you
will need to explore some basic concepts. So what you need to produce for the company is a
short paragraph and the necessary data that would tell other employees how to use an off-theshelf multi meter with a set of standard resistors and a stop watch to measure capacitance.
In-class questions for group discussion: Answer these on the white board when you are
prompted to do so by your instructor. Later the responses of your group will be submitted as part
of your report.
1. What is the definition of capacitance? Don’t look it up, try to come up with it. Discuss it in your
group and try to figure it out. Give a formula and explain what each part means. Also, what
would be the difficulty in trying to measure capacitance by just measuring each quantity in its
definition?
2. At your work station you will be provided with a battery, 2 light bulbs (which are sort of
resistors) 2 switches and a big capacitor. Set up the following circuit and use it to charge and
then discharge the capacitor through the light bulbs. Develop an explanation of what you see.
The teaching assistant will call on one of you to explain.
(+)
3. Exponential decay: (Parts b and c for Phys 212 and 252) Several important processes in
nature are exponential, meaning that when you graph them you see the graph of an exponential
function. One such process is radioactive decay. A certain isotope has a half life of 100 days.
That means if you start with 10 000 nuclei of the isotope at t = 0, then 100 days later you have
(about) 5 000 left, and so on. Starting with 10 000 nuclei, make a graph (with actual numbers, not
just a cartoon) on your white board showing number verses time. Let the time axis on the graph
extend out to two years. Include axis scales and indicate the points where you actually calculated
values using tiny circles.
(a) Copy your sketch here neatly using a ruler.
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RC Circuits
(b) The time constant for any exponential decay is defined using the exponential function. So if y
decays “exponentially in time” then
y  yo e t /
where e is the base of natural logarithms, e = 2.718… or e is about 3. (What is yo ?) This defines
the time constant  : When t increases by one time constant, y decreases to about a third of
what it was. Use this idea to estimate the time constant for radioactive decay of the isotope from
the graph in part a). Explain. Outline your reasoning here.
p
(c) Recall that the log of a product is the sum of the logs, and that the log of x is just p times the
log of x. The natural logarithm
ln  N  of a number is defined in such a way that the natural log
of e is just 1, so
 t
ln  yoe  t /   ln  yo    
 
t

 ln  e   ln  yo  


So if you were to graph natural log of y versus time t, you would get a straight line. What would
the slope and intercept be? Therefore, if you were to plot the natural log of the number of nuclei
surviving at time t versus t, how could you determine the time constant? Discuss this and report
your answers on the white board. Write it here too.
(d) Recall that electric potential, or voltage, is potential energy per charge. That means, for
example, that the unit of potential, i.e. the Volt, is really a Joule/Coulomb. Use dimensions (i.e.
units) to do the following. The fundamental dimensions are length (L), mass (M), time (T) and
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2 -2
charge (Q). Thus for example the dimensions of potential would be Energy/Q = M L T Q .
First, from Ohm’s law, find the dimensions of resistance R. Then from any equation you know,
find the units of capacitance C. Finally, show that RC has units of time. RC is the time constant
in an RC circuit. Thus   RC .
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RC Circuits
Problem 1
You need to find a way to use a stop watch and a standard capacitor to measure the resistance
of a multi-meter used as a volt meter. You may consider the following circuit, where the voltage
source is not really a battery but the DC power supply that you connect to through the wall
terminals.
Measurement plan: Work out a measurement plan, using
the capacitor and stopwatch, for measuring the resistance
of the meter. Tell how to connect the meter, what to do
with the switch, what data to collect, what the group
members will do and so on. Then tell how the data will be
used, graphed, analyzed etc. in order to get the required
value.
V
Record your plan here:
Implementation: Carry out your plan. Collect and record the data here. Also, make notes you
will need later to figure out what the data mean.
Analysis: Show your analysis here and calculate the best value you can for the meter resistance.
Be aware that your employer expects your team to work out the best solution you can, not the
easiest or quickest.
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RC Circuits
Product: Write up a short description of how to measure the resistance of a voltmeter using a
standard capacitance and a stop watch for use by other employees of Ryan Measurement
Technologies. This should be one or two short paragraphs of careful professional writing. Be
sure to include a critical discussion of anything unexpected that might limit the usefulness of this
method of measuring capacitance. Discuss possible sources of error, as usual.
Problem 2
Now you are given some unknown capacitance values to measure. You are to measure them
using a stop watch and the same multi-meter.
Measurement plan: Discuss the measurement plan with your group members. Then record your
measurement plan here. Give all necessary details. (A drawing might be nice.)
Implementation: Do it and record the necessary data here.
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RC Circuits
Analysis: Work out the values of the unknown capacitors here.
calculations. Explain your work succinctly where appropriate.
Show the steps of your
Summary: After discussing with your team members, comment on which parts of the exercises
relate to which of the learning objectives and how they relate.
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