16.3.2
Short and Long Time Behavior of RC Circuits
Kirchhoff’s laws apply to RC circuits (and to any other kind of circuit). To understand an RC circuit, we need
to know how the capacitors affect the potential differences and currents in the circuit. At all times, the potential
difference across a capacitor is by definition
Δ
V
C
=
Q
C
where
Q
is the charge on one plate of the capacitor and
C
is the capacitance. The current “through” the capacitor
is always
I
=
±
dQ
dt
No current actually goes through the capacitor, since charge cannot pass directly from plate to plate.
For a
charging capacitor, equal amounts of negative charge flow away from the positive plate and flow to the negative
plate, making it appear that current is passing directly through the capacitor. Discharging capacitors are discussed
later in the chapter.
+
+
+
_
_
_
I
I
e

e

e

e

Charging Capacitor
If the capacitor is uncharged, the potential difference across the capacitor is
Δ
V
C
=
Q
C
=
0
C
= 0
and it behaves as a wire, an element with no potential difference, in a circuit.
Uncharged Capacitor Behavior:
A fully discharged capacitor has zero potential dif
ference across the capacitor plates and, therefore, behaves as wire in a circuit.
If a capacitor is fully charged, it can accept no more charge, and zero current passes through it. The capacitor
behaves as an open circuit.
Fully Charged Capacitor Behavior:
The current through a fully charged capacitor is
zero; therefore a fully charged capacitor behaves as a break in a circuit.
Example 16.4 Long and Short Time Behavior of a Charging RC Circuit
9
Problem:
In the circuit to the right,
R
1
= 1000Ω
,
R
2
= 1000Ω
,
C
1
= 1000
μ
F
,
C
2
= 2000
μ
F
, and
Δ
V
0
= 10V
.
Initially, all
capacitors are uncharged.
(a)What are the currents and voltage drops across all com
ponents immediately after
S
1
is closed?
(b)What are the currents and voltage drops across all com
ponents after a long time?
Δ
V
0
C
1
C
2
R
1
R
2
S
1
Solution to Part(a)
Immediately after
S
1
closes, the capacitors offer no resistance to the flow of current and so the potential difference
across the capacitors is zero, therefore
Δ
V
C
1
= 0
and
Δ
V
C
2
= 0
.
Because it is in parallel with a capacitor,
Δ
V
R
2
= 0
. Then by Kirchhoff’s Loop Equation,
Δ
V
0
= Δ
V
R
1
= 10V
. The currents in the circuit are found by
Ohm’s Law,
I
1
= Δ
V
1
/R
1
= 10mA
. All this current flows through
C
2
because at the time switch
S
1
is closed,
it presents zero resistance while
R
2
has finite resistance, so
I
R
2
= 0
and
I
C
2
=
I
1
. This current must also flow
through
C
1
, so
I
C
1
=
I
1
.
Solution to Part(b)
After a long time,
C
1
becomes fully charged and blocks all current; the current in all elements of the circuit
becomes zero.
By Ohm’s Law, if zero current is flowing,
Δ
V
R
1
= Δ
V
R
2
= 0
.
This implies
Δ
V
C
2
= 0
.
By
Kirchhoff’s Loop Equation, this means
Δ
V
C
1
= Δ
V
0
= 10V
.
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 Spring '08
 LEWIS
 Capacitance, RC Circuits, Work, Potential difference, Electric charge, Capacitor networks