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 = J_f+ ) (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-m_{oe o}=0;

²E-m_{oe o}=0