Capacitors and Capacitance
P07 -
Capacitors: Store Electric Energy
Capacitor: two isolated conductors with equal and
opposite charges Q and potential difference V
between them.
Q
C
V
Units: Coulombs/Volt
or
Farads
P07 -
Parallel Plate Capacitor
E 0
Q   A
E ?
d
E 0
Q   A
P07 -3
Parallel Plate Capacitor
When you put opposite charges on plates, charges
move to the inner surfaces of the plates to get as
close as possible to charges of the opposite sign
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/electrostatics/35-capacitor/35-capacitor320.html
P07 -4
Calculating E (Gauss’s Law)
o E  dA 
S
qin
0

AGauss
E  AGauss  
0

Q
E 
 0 A 0
Note: We only “consider” a single sheet!
Doesn’t the other sheet matter?
P07 -22
Parallel Plate Capacitor
top
Q
E

dS

Ed

V   
d
A0
bottom
Q

0A

C
V
d
C depends only on geometric factors A and d
P07 -6
Spherical Capacitor
Two concentric spherical shells of radii a and b
What is E?
Gauss’s Law E is not 0 only for a < r < b,
where it looks like a point charge:
E
Q
4 0 r
r̂
2
P07 -26
Capacitance of Earth
For an isolated spherical conductor of radius a:
C  4 0 a
 0  8.8510 F m a  6.410 m
10
40 is roughly 10
F/m
12
6
C  7 10 F  0.7mF
4
A Farad is REALLY BIG! We usually use pF (10-12) or nF (10-9)
P07 -9
1 Farad Capacitor
How much charge?
Q  C V
 1 F 1 2 V 
 12C
P07 -10
PRS Question:
Changing C Dimensions
P07 -30
Energy Stored in Capacitor
P07 -12
Energy To Charge Capacitor
+q
-q
1. Capacitor starts uncharged.
2. Carry +dq from bottom to top.
Now top has charge q = +dq, bottom -dq
3. Repeat
4. Finish when top has charge q = +Q, bottom -Q
P07 -13
Work Done Charging Capacitor
At some point top plate has +q, bottom has –q
Potential difference is V = q / C
Work done lifting another dq is dW = dq V
+q
-q
P07 -14
Work Done Charging Capacitor
So work done to move dq is:
dW  dq V  dq
q
C

1
q dq
C
Total energy to charge to q = Q:
Q
1
W   dW  C  q dq
+q
0
2
1Q

C 2
-q
P07 -15
Energy Stored in Capacitor
Q
Since C 
V
2
Q
1
1
U
 Q V  C V
2
2C 2
2
Where is the energy stored???
P07 -16
Energy Stored in Capacitor
Energy stored in the E field!
Parallel-plate capacitor:

oA
C
and V  Ed
d
U
CV 2  1  o A
2
2 d
1
2

o E  ( Ad )  u  (volume)
E
 Ed  
2
2
2

E
uE  E field energy density  o
2
P07 -17
1 Farad Capacitor - Energy
How much energy?
1
U  C V
2
2
1
2
 1 F 1 2 V 
2
 72 J
Compare to a small capacitor charged to 3kV:
1
U  C | V|2  1 1 0 0 µF  3 k V 2
2
2

1

11 0  4 F
2
31 0 V 
3
2
 450 J
P07 -38
PRS Question:
Changing C Dimensions
Energy Stored
P07 -39