Question

Derive the wave equation in conducting media.

Answer 1

We will start from the Maxwell's equations in the differential form:

(Ampere's Law) ......(1)

(Faraday's Law of electromagnetic induction) .....(2)

(Gauss's law for electric field) ......(3)

(Gauss's Law for magnetic field) ......(4)

where J is current density in the medium, is the resistivity of the medium, D is dielectric field strength, H is magnetic field strength, E is electric field intensity and B is magnetic field intensity.

Now, let us consider a source free uniform medium with magnetic permeability , electric permittivity and conductivity .

Then, the above set of equations would be written as follows:

.....(1.1)

......(2.1)

......(3.1)

......(4.1)

Now, if we take curl on both the sides in the equations (1.1) and (2.1), we get:

and

Now, take one equation at a time using the vector identity :

From the equations (2.1) and (4.1) ,substitute values in the above:

...........(a)

Now, take the equation of double curl on E:

  

From equations (1.1) and (3.1) substitute the values in the above equation:

..............(b)

The equation (a) and (b) are required wave equations defined for conducting medium with conductivity , magnetic permeability   and electric permittivity .