Gauss’s Law
E
q
4 0 R
2
ˆ
R
The beauty of 1/R2
For a point charge,
 E  ds  

q
1 ˆ
R  ds
2
4 0 R
q
q
1
2

R
4

0
4 0 R 2
a constant for any R.
dot product of vectors
For a chunk of charge,
 E  ds  

Q
dV 
0
0
Integral form of
Gauss's law: the
"big picture"
“From our derivation you see that Gauss'
law follows from the fact that the
exponent in Coulomb's law is exactly
two. A 1/r3 field, or any 1/rn field with n
≠ 2, would not give Gauss' law. So Gauss'
law is just an expression, in a different
form, of the Coulomb law...”
-- Richard Feynman
The "big picture"
Gauss's Law
(the "small picture")
with the differential form
The net flux:
(The divergence is a scalar)
This is just notation.
Up to here, just math.
"Gauss's theorem."
0
0
This is the physics
0
0
0
For a dielectric,...
We will talk about what the dielectric constant epsilon really means.
Before that, let's look at some examples in free space (vacuum).
Example: find the field of an infinitely large charge plane
Now, let's talk about dielectrics.
Electric Fields in Insulators (Dielectrics)
Polarization
1. Electronic polarization
d
2. Ionic polarization
p = qd
Net dipole always 0
at 0 external field.
Non-zero net dipole
with external field.
3. Orientational polarization
external charge
--------------------
polarization charge
P
++++++++++++++++++++++ external charge
P  lim
V 0
p
i
i
V
Polarization
(word w/ different meanings
in different contexts)
The field is due to both the "external"
charge and the background charge
There is negative polarization charge at center.
(or polarization charge, described by
the polarization).
P   e 0 E
E 
1
0
 total 
 P    P
1
0
 0  E    P  
(  P ) 
 1
 P
0 0
polarization charge
"external" charge
  ( 0   e 0 )E    D   ,
where    0 (1   e )   0 r , D   0 r E  E
By doing this, we just lump the polarization (or contribution by the polarization
charge) into the material property -- the dielectric constant.