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Capacitors and Capacitance
Capacitors are devices that can store a charge Q at some voltage V.
The greater the capacitance, the more charge that can be stored. The
equation for capacitance, C, is very simple:
C
Q C Q
 F


V V V
1F  1 farad
Even an isolated conducting sphere has capacitance.
We just take the equation for the electric potential of a
point charge (which also applies to conducting
spheres, and find the above ratio:
Q
R
V
kQ
R
C
Q R

V k
Michael Faraday
Find the capacitance of “Old Sparky”
r
0.20m
12
C 

22
.
2

10
F  22.2 pF
9
2
2
k 9.00 10 N  m / C
A capacitance of ~ 20 pF is not very big.
If this Van de Graaf generator could reach
1,000,000 V, the total charge on the dome
would be only 20 mC. This is one reason
this device is safe. The other is that high
voltage causes charge to travel along the
surface of objects (such as our bodies).
The generic capacitor
It is more practical to build a
conductor out of two
conductors, charging them
equal and opposite, as an
electric dipole. With the right
geometry, this will create a
“contained” electric field, and
the possibility of much larger
capacitance than can be
achieved with a sphere of
equivalent size.
Easily calculable capacitor #1: parallel plates
Here’s the example we’ve been referring to for some time: the parallel
plate capacitor. We assume that the plates are so large compared to
their separation d that we can ignore the fringe field, and that all the
electric field is inside. Then, as we found earlier, V = Ed. And, since
all the charge is on the inner surfaces of the conductors, E = s /e0 ‘
Q sA sA  e 0  e 0 A
C 

 
V Ed
d s 
d
What happens to C if d
is made very small???
This is a simple result. Notice that C
depends only on e0 and the dimensions
of the capacitor. We will see the same
thing with all other capacitors.
Calculable capacitor #2: concentric spheres
This calculation is also easy. We use
the equation for the electric potential of
a sphere of a given radius and charge.
We apply it once to the inner sphere,
and again to the outer, to find the
voltage difference between the two.
(By Gauss’s Law, the only electric field
is between the two spheres.)
 rb  ra 
kQ kQ

V  Va  Vb 

 kQ
ra
rb
 ra rb 
Then, the capacitance of
a spherical capacitor (in
vacuum) is:
Q 1  ra rb  4e0ra rb
 
C
 
V k  rb  ra  rb  ra
Calculable capacitor #3: coaxial infinite cylinders
We’ve done most of the work
already for this, by finding the
difference in electric potential at
two different radii in a cylindrically
symmetric geometry. We simply
use that earlier result for V, and
find C from the basic equation.
 rb
V  2k ln 
 ra



Notice that to calculate a Q for the capacitance equation, we need to
choose a length, L, so that Q = L:
Q
L
L
C
1
2e0 Capacitance


C


L
 rb 
 rb  per unit length
V
 rb 
 rb 
2k ln   ln   (in vacuum).
2k ln   2k ln  
 ra 
 ra 
 ra 
 ra 
Popular design for early capacitors – the Leyden Jar
In 1745, the Leyden Jar (or Leyden Bottle)
was invented by Ewald Jürgen von Kleist
(1700-1748). The glass increases the
capacitance dramatically, as we shall
understand soon …
Glass
A “Leyden Battery”. What would we
call this configuration today? Again,
we’ll be discussing this further …
Early charging machines
Induction machine invented by James
Wimshurst c. 1880. The two disks, with
narrow tin-foil strips glued around the rim,
are rotated in opposite directions by a
system of pulleys and belts. At each side
there is a conductor, terminating in metal
brushes that rub against the tin-foil sectors.
The charge is induced in two jaw collectors
and stored in a pair of Leyden jars
connected to two sliding electrodes. This
machine was highly popular and is still
used today for teaching purposes. It
worked well even in damp weather and did
not reverse polarity. Apart from laboratory
demonstrations, it was used for medical
treatment and as a high-voltage source for
the first X-ray tubes.
By ~ 1905. Large machines!
Modern capacitors: many sizes, shapes, and types
http://en.wikipedia.org/wiki/Capacitor
http://www.sparkmuseum.com/RADIOS.HTM
http://www.uoguelph.ca/~antoon/gadgets/caps/caps.html
Capacitor types:
(most common types are in red).
Metal film: Made from polymer foil with a layer of metal deposited on surface. They have good quality and
stability, and are suitable for timer circuits and high frequencies.
Mica: Similar to metal film. Often high voltage. Suitable for high frequencies.
Paper: Used for high voltages.
Glass: Used for high voltages. Stable temperature coefficient in a wide range of temperatures.
Ceramic: Chips of altering layers of metal and ceramic. Very common, they find use in low-precision
coupling and filtering applications. Good for high frequencies.
Electrolytic: Polarized. Similar to metal film in construction, but the electrodes are made of aluminum
etched for much higher surface area, and the dielectric is soaked with liquid electrolyte. Can achieve high
capacities.
Tantalum: Like electrolytic. Polarized. Better performance at higher frequencies. Can tolerate low
temperatures.
Supercapacitors: Made from carbon aerogel, carbon nanotubes, or highly porous electrode materials.
Extremely high capacity.
More discussion to come on the subject of dielectrics…
Capacitors connected in parallel
Consider starting with
uncharged capacitors, then
connecting a battery across the
terminals. What is the total
charge in this circuit? Why do
the two capacitors have the
same voltage, in equilibrium?
Use C = Q/V …
Solve for the total capacitance of
this equivalent circuit, Ceq, in terms
of C1 and C2.
 Derive the equation
Capacitors connected in series
Consider starting with uncharged
capacitors, then placing a charge
+Q on capacitor C1. Why do the
two capacitors have the same
charge, in equilibrium? What is the
relationship among the voltages?
Use C = Q/V …
Again, solve for the total
capacitance of this “equivalent
circuit”, Ceq, in terms of C1 and C2.
 Derive the equation
Equations for capacitors connected
in parallel and in series
Parallel:
C  C1  C2  ...
Series:
1
1
1


 ...
C C1 C 2
Finding the equivalent capacitance of a complex
circuit by successive combination of its elements,
using the equations we just derived.
These circuits are purely capacitive. No resistors, etc.
 Do
Potential energy stored in a capacitor
+Q
A
We will calculate the work required
+
+
+
+
to start with a discharged capacitor
and charge it to a total charge Q.
Imagine we are taking positive
V
dq
d
E
charge from the lower plate in
_
_
_
_
increments dq and moving them
through the voltage difference V
-Q
created by the electric field. The
Q
q
1
work required for this move is qV.
U  dW  Vdq 
dq 
qdq 
With each dq that is moved, V
C
C0
increases, so that the work for
This is the potential energy in any
each dq rises as the capacitor
charged capacitor since the
charges. We are describing the
derivation is general!
integration at right, with the final
For the case of parallel plates:
answer being the potential energy
stored in the capacitor.
2


C

e0 A
d

U
Qd
2e 0 A
+
_
Q2
2C
General expressions for potential energy in a capacitor,
and energy density of the electric field.
There are two other forms of the general expression for the potential
energy in a capacitor, derivable from the expression on the last page by
using V = Q/C to eliminate one of the three variables, V, Q, or C, from
the equation:
2
2
Q
CV
QV
U


2C
2
2
Where is the energy stored? Amazingly, in the electric field itself! For
the parallel plate capacitor, which has a constant electric field, it is easy
to calculate the energy in this field, starting from the second form
above:
1
1e A
1
U  CV 2   0 ( Ed ) 2  e 0 E 2 ( Ad )
2
2 d 
2
The factor (Ad) is the volume of this capacitor. If we
divide by this factor, the result will be the energy
density, a general formula that applies (point by point)
to all electric fields in a vacuum:
1
u  e0E 2
2
Charging one capacitor from another:
final conditions, and energy.
 Do
Why have we been saying “in a vacuum”?
If we fill the volume of any capacitor with a “dielectric” material,
we will see the following:
C0
C
For the same charge Q, the voltage on the capacitor in
vacuum, V0, will be greater than the voltage, V, on the same
capacitor filled with dielectric. Since C = Q/V, the
capacitance has been increased by V0 /V.
Very useful, but how does this happen? 
What’s a “dielectric material”? It’s an insulator.
And, the property of a dielectric that causes the increase of
capacitance is its “polarizability.” There are two classes of
dielectrics, with different polarization mechanisms:
Polar molecules
Partial alignment with E
Non-polar molecules
E induces polarization in
each atom or molecule
In both cases, each dipole has an interior electric field that
points opposite to the applied field E. So, the total electric
field is reduced. Then, since V in a capacitor is the integral
of E with distance, V is reduced.
The “dielectric constant”, K, is the factor by
which C increases: C = KC0
Dielectric constants for various materials:
More dielectric constants, plus some “breakdown voltages”:
Dielectrics in capacitors: bringing all the physics together
First, consider a block of dielectric in an external field E.
Essentially, the block is still net neutral inside, but the
polarization has induced two surface charge densities, / si ,
with the negative surface charge on the “incoming” face, and
the positive on the outgoing.
Using the parallel plate capacitor as an example, putting this
dielectric in the gap reduces the electric field as follows:
K
C
Q / V V0 V0 / d E0




C0 Q / V0 V
V /d
E
E
The surface charges determine E0 and E:
s
s si
E0 
E
e0
e0
Put these into the top equation to find si:
s si 1 s

e0
K e0


s i  s 1 
1

K
If K is very large, si is approximatelys !
E0
K
(K > 1)
E0
E
Dielectrics in capacitors, continued…
Recall that e0 is called the “permittivity of the vacuum”. We can
imagine that the polarization of a dielectric changes this factor. So we
define a permittivity for the material, e = Ke0 that takes the
polarization into account. Then we can modify the vacuum equations
to apply to cases with dielectrics, simply by writing e in place of e0 .
Showing how this works for a parallel plate capacitor:
 e0 A
 A  eA
C  KC0  K 
  Ke 0   
d  d
 d 
U
1
1
1
 1
CV 2  ( KC0 )V 2  K  e 0 E 2 ( Ad )  eE 2 ( Ad )
2
2
2
 2
C
eA
d
1 2
u  eE
2
We have illustrated this simple replacement e0  e for a parallel plate
capacitor, but it applied to all capacitors, and to all equations involving
permittivity.
Using Gauss’s Law with dielectrics
  Qencl  free
 E   KE  dA 
A
e0
We were actually using this law
when we calculated the electric
field in the dielectric based on the
sum of surface charges, and
found the equation below.
s si
E
e0
Energy in a capacitor with and without dielectric:
It depends on what we hold constant!
Constant Q: charge
and disconnect.
+Q
K
U before
-Q
Q2

2C
U after
Q2

2 KC
K>1  U is smaller with the dielectric is inside the capacitor.
Constant V: leave
connected to battery.
K
U before 
V
1
CV 2
2
U after 
1
KCV 2
2
K>1  U is larger with the dielectric is inside the capacitor.
Are there forces on these slabs as they are being inserted?
Examples with dielectrics
Discuss
More examples with dielectrics
Discuss
Examples of series/parallel capacitor circuits
Example with a switch: Series?
Parallel? Neither?
Discuss
Beware: these are not simply parallel/series connections!