### capacitors

```Capacitors and Dielectrics
Capacitors
•Conductors are commonly used as places to store charge
–Q
•You can’t just “create” some positive charge somewhere, you
have to have corresponding negative charge somewhere else
+Q
Definition of a capacitor:
b
a
V
•Two conductors, one of which stores charge +Q, and the
other of which stores charge –Q.
Can we relate the charge Q that develops to the voltage difference V?
k Q
•Gauss’s Law tells us the electric field between the conductors:
E  e 2 rˆ
•Integration tells us the potential difference
r
a
V   
b
keQ
r
2
dr 
k eQ
r
a
b
1 1
 V  k eQ   
a b
Capacitance
•The relationship between voltage difference and charge is
1 1
 V  k eQ   
normally linear
a b
Q  C V
•This allows us to define capacitance
•Capacitance has units of Coulomb/Volt
C
F
•Also known as a Farad, abbreviated F
V
•A Farad is a very large amount of capacitance
•Let’s work it out for concentric conducting spheres:
C 
Q
V

ab
What’s the capacitance of the Earth, if we
put the “other part” of the charge at infinity?
ke b  a 
a  6370 km
b
C 
ab
ke b  a 

ab
keb
6.37  10 m
6

8.988  10 N  m / C
9
2
2

7.09  10
4
N m / C
2
 709  F
Parallel Plate Capacitors
•A “more typical” geometry is two large, closely spaced, parallel conducting plates
•Area A, separation d.
Let’s find the capacitance:
•Charge will all accumulate on the inner surface
A
•Let + and – be the charges on each surface Q   A

•As we already showed using Gauss’s law, this
E
nˆ
means there will be an electric field given by:
0
•If you integrate the electric field over the distance d,
you get the potential difference
d
V    E  ds 
C 
Q
V
C 


0A
d
0
dx 
d
0

Qd
0A
To get a large capacitance, make
the area large and the spacing small
Circuit symbol
for a capacitor:
Capacitors in Parallel
•When capacitors are joined at both ends like
this, they are said to be in parallel
•They have the same voltage across them
•They can be treated like a single capacitor:
Q1  C 1  V
Q2  C 2 V
V
C1
C2
Q  C V
Q  Q1  Q 2   C 1  C 2   V
C  C1  C 2
Capacitors in Series
•When capacitors are joined at one end, with
nothing else, they are said to be in series
•They have the same voltage across them
•They can be treated like a single capacitor:
Q  C 1  V1
Q  C 2  V2
 V   V1   V 2 
Q
C1

Q
C2
V
C1
C2

Q
1
C
C

1
C1

1
C2
Series and Parallel
•When two circuit elements are connected at one end, and nothing else is
connected there, they are said to be in series
1
C1
C2

C
1

C1
1
C2
•When two circuit elements are connected at both ends, they are said to
be in parallel
C1
C2
C  C1  C 2
•These formulas work for more than two circuit elements as well.
C4
C2
C3
1
1
1
1
1
1
C1

C5
C

C1

C2

C3

C4
C5
Complicated Capacitor Circuits
•For complex combinations of capacitors, you can replace small structures by
equivalent capacitors, eventually simplifying everything
4
3
The capacitance of the capacitors in pF
2
6 5
1
at right are marked. What is the effective
2
capacitance of all the capacitors shown?
10 V
•Capacitors 1 and 5 are connected at C  1  5  6
both ends- therefore they are parallel
•Capacitors 3 and 6 are connected at just
one end – therefore they are series
1
C

1
3

1
6

1
C  2
2
•All three capacitors are now connected at both ends – they are all in parallel
C  2428
Energy in a Capacitor
•Suppose you have a capacitor with charge q already on it,
and you try to add a small additional charge dq to it, where
dq is small. How much energy would this take?
•The side with +q has a higher potential
•Moving the charge there takes energy
•The small change in energy is: d U   d q   V   dq  q
q  C V
–q
+q
C
•It makes sense to say an uncharged capacitor has U = 0
q Q
U 

q0
U 
Q
Q
dU


0
q dq
C

q
2C
2
2C
2
Q  C V
Q

0
Q
dq
2
2C
U 
Q
2
2C

C  V
2

2
Energy density in a capacitor
Suppose you have a parallel plate capacitor with
area A, separation d, and charged to voltage V.
(1) What’s the energy divided by the volume between the plates?
(2) Write this in terms of the electric field magnitude
U 
1
2
C  V

2

0A
2d
 V 
A
2
•Energy density is energy over volume
u 
U
V

U

 0 A  V
2

2
 V 
 0 

2
 d 
1
u 
1
2
2
0 E
E 
V
d
2
•We can associate the energy with the electric field itself
•This formula can be shown to be completely generalizable
•It has nothing in particular to do with capacitors
d
Dielectrics in Capacitors
•What should I put between the metal plates of a capacitor?
•Goal – make the capacitance large
•The closer you put the plates together, the
 A
bigger the capacitance
C  0
•It’s hard to put things close together –
d
unless you put something between them
•When they get charged, they are also very
attracted to each other
•Placing an insulating material – a dielectric –
allows you to place them very close together
 0 A
C 
•The charges in the dielectric will also shift
d
•This partly cancels the electric field
•Small field means smaller potential difference   1
•C = Q/V, so C gets bigger too
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
–
–
–
–
–
–
–
–
–
–
–
–
Choosing a dielectric
What makes a good dielectric?
•Have a high dielectric constant 
•The combination 0 is also called , the permittivity
•Must be a good insulator
•Otherwise charge will slowly bleed away
•Have a high dielectric strength
•The maximum electric field at which the insulator
suddenly (catastrophically) becomes a conductor
•There is a corresponding breakdown voltage
where the capacitor fails
C 
 0 A
d
What are capacitors good for?
•They store energy
•The energy stored is not extremely large, and it tends to leak away over time
•Gasoline or fuel cells are better for this purpose
•They can release their energy very quickly
•Camera flashes, defibrillators, research uses
•They resist changes in voltage
•Power supplies for electronic devices, etc.
•They can be used for timing, frequency filtering, etc.
•In conjunction with other parts
Dipoles
•We’ve done a lot with charges in electric fields
•However, in nature, neutral combinations are much more
common than charged objects
•This doesn’t mean there are no electric effects!
•A dipole is any collection of 2+ charges that have no total
charge, but the charge is lopsided on one side or the other
•Many molecules are dipoles
•The dipole moment for a pair of charges, is just a vector
equal to the charge q times the separation vector r
•For more complicated objects, it is harder
p
+q
-q
r
qr
i i
i
p  qr
p
  r dV
C+ O-
H+
H+
O-
Dipoles in Uniform Electric Fields
•Electric fields are often uniform, or nearly uniform, as seen by a molecule
•After all, molecules are pretty small!
•A dipole in a uniform electric field feels no total force p   q i ri
i
•However, there is a torque, or twisting force, on a dipole
F
F
i

 q E  E q
i
i
τ
i
0
r
i
r F
i
i
+q

i
r qE
i
i
i


   q i ri   E
 i

F
-q
i
τ  pE
F
τ  p E sin 
•There is also an energy associated with a dipole in an electric field
U 
V q
i
i
i
   q i E  ri   E   q i ri
i
i
U  E  p
U   E p co s 
Dipoles as Dielectrics
•In the absence of an electric field, dipoles will orient randomly in different
directions due to random thermal motion
•When you turn the electric field on, the
random motions still continue, but a
fraction of the molecules will reorient to
match the electric field
•Now there is an excess of positive charge
on the right and negative charge on the left
•This creates a weaker counter-balancing
electric field that partly cancels the
imposed field
```