Capacitance and Dielectrics
Capacitors and Capacitance
In mechanics we are used to devices which store potential energy
Is there a way to store electric potential energy
Capacitors
Any 2 conductors insulated from each other form a capacitor
can be realized by an insulating material (dielectric) or vacuum
In circuit diagrams a capacitor is represented by
the symbol:
Let’s charge a capacitor
-Q
+Q
Vab=Va-Vb
Vb
Va
Remember our electric field calculations for various charged objects
E ( x, y  0) 
x Q
ex
4 0 r 3

Q 
E  ex
1
2
2 R  0 



2
 R / x   1 
1
We always find E  Q
b

Since Va  Vb  E d r
voltage Vab  Va  Vb  Q
a
Capacitance C 
SI unit 1F=1C/V
Q
Vab
depends only on geometry and dielectric properties,
not on charge
read F=Farad in honor of Michael Faraday
Capacitance is an intuitive characterization of a capacitor. It tells you:
how much charge a capacitor can hold for a given voltage (potential difference).
The more the higher the capacitance
Calculating capacitance
According to C 
Charge density
 Q/ A
+++++++++++++++++++++++++
Parallel-plate capacitor
the major task in calculating C is calculating Vab
-------------------------------------------
Q
Vab
Homogeneous field, E=/0 for
the limit d<< plate dimensions
Using
b
Va  Vb   E d r
a
Vab  Ed
 d Qd


0
A 0
d
We obtain the capacitance C of a parallel-plate
capacitor in vacuum as
A
C
0
d
Note: 1F is a huge capacitance. More typical values are 1µF=10-6F to 1pF=10-12F
Demonstration: parallel-plate capacitor
A few slightly more involved examples
Capacitance of a spherical capacitor
Step 1:
calculate the electric field using Gauss’s law between the 2 spheres
 E dA
Q
0
Image from
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capsph.html
Q
E 4 r 
Q 0
E
4 0 r 2
2
for
arb
Step 2: calculate the voltage Vab for a given amount of charge Q on the spheres
Qdr
Q 1

2
4

r
4 0 r
0
a
b
b
Vab  
Q
Step 3: applying C 
Vab

a
C
Q 1 1
  
4 0  a b 
4 0 4 0ab

1 1
ba

a b
Clicker question
Does an isolated (individual) charged sphere have capacitance?
1) No, where would the electric field lines end?
2) Yes, I just don’t know the value
4 0
3) Yes, it is a special case of C 
1 1

a b
in the limit b->
Image from
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/capsph.html
Capacitance per length of a cylindrical capacitor (coaxial cable)
Step 1:
calculate the electric field using Gauss’s law
Q / L
E
dA
Q
0
Cylinder with a<r<b
L
 Q/L
L
E 2 rL 
0

E
2 r 0
Step 2: calculate the voltage Vab

b
 dr
Vab  

ln
2 0 r
2 0 a
a
b
C
Q


Step 3: applying 
L LVab Vab
C 2 0

L ln b
a
C
1
 55.6 pF / m
b
L
ln
a
Typical value for antennas, VCRs 69pF/m
Clicker question
How did I like the first midterm exam?
A) I have a thing for midterms, this one rocked as usually*
B) The midterm was hard and unfair.
C) The midterm was as expected
D) I hate exams and this one was particularly bad. Hate it, hate it, hate it.
*