Buggé/Sierzega: Circuits 3
Energy Stored In the Electric Field of a Capacitor
A capacitor has potential energy stored in it. The amount of potential energy can be
thought of as equal to the amount of work needed to separate the charge in the first place.
Remembering the relationship between work & potential difference (from the previous
topic), we can say that the amount of work needed to transfer a charge dq' is given by
dW = V dq' = (q'/C) dq'
where q' is the amount of charge that has already been transferred to the capacitor. And
q’/C is the voltage across the capacitor at any given moment during the interval of time
during which the total charge is transferred. (Charge is not transferred all at once; it builds
up.)
Q
W =  dW = (q'/C) dq'
0

W = UC = (1/2) Q2 /C
Since Q = CV, we also have that
UC = (1/2) CV2
&
UC = (1/2) QV
Example 1 - Using the Example of a Series/Parallel arrangement of capacitors from
Circuits 2, find
8F
(a) the total stored energy
(b) the energy stored in the 6F capacitor
(c) the energy stored in the 8F capacitor
4F
6F
12 V
Buggé/Sierzega: Circuits 3
Example 2 - Use the capacitor "circuit" shown below to find the following:
12 F
a) Ctotal
b) Qtotal
6V
c) Determine the charge on each capacitor
10 F
15 F
H int: The 12F is not in parallel w ith the 10F
because their (-) plates aren’t connected!!!
d) V10 F
e) U12 F
f) U15 F
(18 F ; 108 C ; 72 C, 36 C, 36 C ; 3.6 V ; 216J ; ~ 43 J )
Example 3 - Use the arrangement shown below to find:
a) Ctotal
8F
12 F
b) Qtotal
4F
3V
3F
c) Q on each capacitor
6F
3F
Buggé/Sierzega: Circuits 3
Conservation of Charge & Electrostatic Equilibrium
We can extend our analysis of arrangements of capacitors by employing two more
principles (Conservation of Charge & Electrostatic Equilibrium) along with those already
established for series & parallel capacitors.
When do we do this? Here’s the typical situation (illustrated below): a capacitor C1 is
charged by a battery. The battery is then removed and the charged capacitor C1 is
connected to one or two uncharged capacitors.
S
C1
C2
V
Heads up! If there are 2 uncharged capacitors that are in series or parallel,
as shown below, then C2 is the equivalent capacitance of those 2 uncharged
capacitors.
S
V
C1
S
C2
V
C1
C2
The following two principles apply to these situations:
1. Conservation of Charge: The total amount of charge Q0 before and after
the switch is closed stays the same.
Q0 = Q1 + Q2
2. Electrostatic Equilibrium : After the switch is closed, the charge will quickly
“rearrange” itself in such a way that it will subsequently remain static.
In other words, the final potential difference, Vf , across both C1 and C2 must be the same
(which is always true for parallel capacitors anyway).
Q1/C1 = Q2/C2
Buggé/Sierzega: Circuits 3
Example 4 – In the arrangement shown below, Switch S1 is closed while switch S2 remains
open.
(a)
Once equilibrium is established,
i. Determine the charge on the 5 F capacitor
ii. Determine the energy stored in the 5 F capacitor
Next, Switch S1 is opened and switch S2 is closed. Equilibrium is once again established.
(b)
Determine the charge on both capacitors
(c)
Determine the energy stored in both capacitors
(d)
Determine the potential difference across both capacitors.
(e)
Compare the stored energy both before and after switch S2 is closed.
Account for any differences.
S1
50 V
S2
5F
10 F
Buggé/Sierzega: Circuits 3
Predict & Test
1. Use the online simulation found at http://phet.colorado.edu/en/simulation/circuitconstruction-kit-ac to construct the following circuits.
2. For each of the arrangem ents show n below , u se the pow er supply and voltm eter to
charge the capacitor on the left sid e of each arrangem ent to a potential d ifference of 3
V. Be sure to follow the polarity as shown!!!
Then, d isconnect the pow er supply.
This is the capacitor on the left side of each
of the arrangements show n below .
3V
a. When sw itch S is closed , Predict & Test the final voltage across the other
capacitors in the arrangem ent.
b. Choose any one of the arrangements, and compare the initial stored energy to
the final stored energy. Account for any differences.
(a)
S
3V
(b)
330 F
100 F
S
330 F
3V
470 F
100 F
Buggé/Sierzega: Circuits 3
(c)
S
3V
470 F
330 F
100 F