in linear, homogeneous, isotropic
medium at rest
Maxwell’s Equations:
DivE=×
E= (Gauss’s electric law)
or ×
D= (1)
DivB=×
B=0 (no name) (2)
Curl E=xE
= - (Faraday’s Law)
(3)
Curl B=xB
=m J+me
(Ampere’s Law corrected by Maxwell) (or xH
= Jf+ )
(4)
Use Ohm’s law: J =s E
s - conductivity
Apply curl to (4):
x (xB) = x( ms E )+ x (me)
Interchange the space and time derivatives:
x (xB)
=ms (
xE )+ me
Use Faraday’s law:
x (xB) =-ms - me
Use
x (xB)= (× B)- 2B
where 2 is Laplacian
Remember that × B=0
Finally obtain:
2B-ms-me=0
For the electric field:
Apply curl to
xE = -
x(xE) = - x()=-
x(xE)= (× E)- 2E=-=
= - (mJ+me)=-(mFE+me)
Use × E= and Ohm’s Law.
Finally obtain :
2E-mF-me=r /e
EM wave equations:
2B-ms-me=0
2E-ms-me=r
/e
For non-conducting media:
s =0
2B- me=0;
no chages r =0
2E-me=0
In vacuum:
e =e0=8.85x10-12C2/N-m2
m =m0=4p
x10-7N/A2
2B-moe o=0;
2E-moe
o=0