RC Circuits
- circuits in which the currents vary in time
- rate of charging a cap depends on C and R of circuit
- differential equations
Quiz:
After the switch is closed, the light from the bulb:
A) Is brightest just after the switch is closed, then
fades slowly and disappears.
B) Increases gradually to maximum brightness, over a
few seconds, then remains steady.
C) Comes to maximum brightness immediately, and
remains steady.
D) The bulb remains off.
10 
0.2 F
Discharging a Capacitor
I
C
1)
2)
q
Given: R, C, qo (initial charge)
R
-q
q
 IR  0
C
I 
dq
dt
Find: q(t) and I(t) when switch
is closed
(Kirchhoff’s Loop Rule)
(- sign because q decreases for I > 0
That is, current in circuit equals the
decrease of charge on the capacitor)
Combine 1) and 2) to get:
I
C
q
-q
R
dq
dt

q
RC
where: q = q(t)
q(0) = qo
This is a differential equation for the function
q(t), subject to the initial condition q(0) = q0 .
We are looking for a function which is proportional
to its own first derivative (since dq/dt ~ -q).
Solution:
q(t)  q ο e

t
RC
RC is called the “time constant” or “characteristic
time” of the circuit.
Units: 1 Ω x 1 F = 1 second (show this!)
Write t (“tau”) = RC, then:
q (t )  q o e
t 
 
t 
(discharging)
Discharging
q
qo
q (t )  q o e
t
 t
t
t  RC
2t
3t
t ,
t = 2 t,
t = 3 t,
q ≈ 0.37 qo
= (qo/e)
q ≈ 0.14 qo
= (qo/e2)
q ≈ 0.05 qo
= (qo/e3)
t∞,
q0
= (qo/e∞)
t=
t
Draw a graph for I(t).
Quiz
A capacitor is charged up to 18 volts, and then
connected across a resistor. After 10 seconds,
the capacitor voltage has fallen to 12 volts.
Find the time constant RC, and…
What will the voltage be after another 10
seconds (20 seconds total)?
A)
B)
C)
D)
8V
6V
4V
0
Charging a capacitor

C is initially uncharged, and the switch
is closed at t=0. After a long time,
the capacitor has charge Qf .
C
R
 R
Then,
dq
dt

q
0
C


q(t)  Q f  1 - e

t
t


where t  RC.
Question: What is Qf equal to?
Charging a capacitor


q (t )  Q f  1  e

t  RC
q
t
t


Qf
t
t = 0,
q=0
t = RC,
q 0.63 Qf
t = 2 RC, q 0.86 Qf
2t
3t
t = 3 RC, q 0.95 Qf
etc.
t
Draw a graph of I(t). Why is I=+dq/dt this time?
Example 2
100 kΩ
12 V
2 µF
The capacitor is initially uncharged.
After the switch is closed, find:
i) Initial current
ii) Initial voltage across the resistor
iii) Initial voltage across the capacitor
iv) Time for voltage across C to reach 0.63*12V
v)
Final voltage across the resistor
vi) Final voltage across the capacitor
Solution
Example:
A 2kΩ and a 3kΩ resistors connected in parallel are
connected in series with a 2uF and a 3uF capacitors
that are connected in parallel. The power source is 120V.
Find the charge on each capacitor as a function of time.
Quiz:
In a simple circuit with a capacitor, resistor and a switch,
long time after the switch is closed, the current in the circuit
will be:
A) ε/R
B) ε/τR
C) ε/eR
D) zero
“RC” Circuits
• a capacitor takes time to charge or discharge
through a resistor
• “time constant” or “characteristic time”
t
= RC
(1 ohm) x (1 farad) = 1 second