Assume a Uniform, Homogeneous,linear,isotropic and Stationary medium with Non-zero current .
When an EM wave is travelling in a conducting medium in which . The wave is rapidly attenuated in a conducting medium and in a good conductors, the attenuation is so high at Radio frequencies. The wave penetrates the conductor only to a small depth.
choose the equation the time-domain representation of it is
since in phasor-notation , and .
By differentiating the above equation with respect to time
From the Maxwell Equation
By taking curl on both sides
By using the vector identity
From Equation (2)
from Maxwell’s equation
This is called Wave equation (Electric field)for a general medium when .
wave equation for Magnetic field of a general medium is
Case 1:- wave equation for a conducting medium
for a conductor the net charge inside an isolated conductor is , then the wave equation for a conducting medium () is
The above equations are known as wave equations for conducting medium and are involving first and second order time derivatives, which are well known equations for damped(or) attenuated waves in absorbing medium of homogeneous, isotropic such as metallic conductor.
Case 2:- Wave equation for free space/ Non-conducting medium/loss-less medium/Perfect Di-electric medium
The conditions of free space are and
By substituting the above equations in general wave equations, the resulting wave equations for non-conducting medium are