Physics 9
WS E8 (rev. 1.2)
Page 1
E-8. RC circuits
Questions for discussion
1.
2.
Why does the current vary with time in an RC circuit?
For the simplest type of RC circuit (consisting only of a battery ε , a resistance R, and a capacitance C), the time
constant is τ = RC. What is the significance of this time constant? In other words, what does the time constant
for an RC circuit tell you?
Physics 9
Page 2
WS E8 (rev. 1.2)
Problems
c)
1.
a)
In the RC circuit shown below, the capacitor is
initially uncharged, and the switch is initially open.
Immediately after the switch is closed, what is the
voltage across the capacitor?
b) Immediately after the switch is closed, what current
flows in the circuit? (Hint: You may want to redraw an equivalent circuit, based on your answer
from part (a). Then use the Loop Rule.)
2.
a)
In the RC circuit shown below, the capacitor carries
an initial charge q0. The switch is initially open.
What will happen when the switch is closed?
b) Immediately after the switch is closed, how much
current flows in the circuit? (Hint: Loop Rule.)
3.
After a very long time, what current flows in the
circuit? Why?
d) After a very long time, what is the voltage across the
capacitor? (Hint: Loop Rule, taking into account
your answer from part (c).)
e)
After a long time, how much charge has accumulated
on the capacitor plates?
f)
Sketch a graph showing the charge on the capacitor
plates as a function of time. Let t=0 be the instant
the switch was closed.
g)
Sketch a graph showing the current in the circuit as a
function of time. Again, let t=0 be the instant the
switch was closed.
h) How are these two graphs related? Can you explain
why the graphs are related in this way? 
c)
After a long time, how much current flows in the
circuit? Why?
d) After a long time, what is the voltage across the
capacitor? How about the charge on the capacitor?
e)
Sketch graphs showing the charge on the capacitor
and the current flowing in the circuit as functions of
time.
f)
How much energy was stored in the electric field of
the capacitor initially?
g)
How much energy is stored in the electric field of the
capacitor after a long time?
h) What happened to this energy? 
In the RC circuit shown below, the capacitor is
initially uncharged, and the switch is open.
a)
Then, at t=0, the switch is closed:
Immediately after the switch is closed, what are the
currents i1, i2, and i3?
b) After a long time, what are the currents i1, i2, and i3?
c) After a long time, how much charge is on the
capacitor plates? 
Physics 9
4.
Page 3
WS E8 (rev. 1.2)
Consider once again the RC circuit from Problem 2.
Initially the capacitor carries a charge q0, and the
switch is open.
b) What is the relationship between the current i(t) and
the charge q(t) contained on the plates? Use this to
express your Loop Rule entirely in terms of q, dq/dt,
and constants.
c)
Verify that the function q(t) = q0e-t/RC satisfies the
Loop Rule at all times. Hence, this function gives
the charge on the capacitor at any given time after the
switch is closed.
d) Sketch a graph of the charge on the capacitor as a
function of time.
When the switch is closed at t=0, charge will begin
to leak off of the capacitor plates, resulting in current
flow around the circuit. At some arbitrary time t, the
circuit therefore looks like this:
a)
Write down the Loop Rule for this circuit.
e)
From the expression q(t) = q0e-t/RC, find an expression
for i(t), the current flowing in the circuit at any time
after the switch is closed. Sketch a graph of this
function.
f)
Verify that these graphs agree with the ones you drew
for part (e) of Problem 2.
g)
From your expression for i(t), find an expression for
P(t), the rate of heat loss through the resistor at any
given time after the switch is closed.
h) By integrating P(t) over time, from t=0 to t= ∞ ,
show that the total amount of heat dissipated by the
resistor is none other than the initial energy stored in
the capacitor, as required by energy conservation.
(This justifies your answer to part (h) of Problem 2.)
