EM waves in conducting medium (medium with finite μ, ε and σ)

EM WAVES IN CONDUCTING MEDIUM

(Medium with Finite μ, ε and σ)

General Maxwell's equations are

In conducting medium σ ≠ 0 i.e., there is a conduction current in the medium therefore   but the charge density is zero everywhere (inside the conductor the charge density is always zero). That is ρ = 0.

Therefore the eqn. (1) reduces to

or

Here and ε is permittivity of the medium. 

Taking the curl on both sides of eqn. (3), we get

From vector calculus identity, we have

But from eqn.

Therefore, equation (7) becomes

Substituting the eqn. (8) and (9) in (6), we get

On substituting the value of from eqn. (4) in eqn. (10), we have

Since equation (11) becomes

This is the general wave equation for the electric vector in an electromagnetic wave propagating in conducting medium.

In a similar way, by taking the curl of the eqn. (4) we obtain the general wave equation for the magnetic vector in a conducting medium as

Wave Equation for Plane Polarized EM Waves

Let us consider that the electromagnetic wave is travelling in the x-direction and the electric vector is directed along the y-axis and the magnetic vector is directed along the z-axis. For such a wave we have

Therefore, the wave equations from (12) and (13), reduce to

In the above wave equations where v is the velocity of electromagnetic wave in conducting medium. The product μσ is called magnetic diffusivity.

Thus, the finite conductivity adds the diffusion term to the wave equation. This is due to the presence of conduction current

For vacuum or perfect insulators (σ = 0) and so these equations reduce to the expressions corresponding to the free space or dielectric medium.

Solution of the plane Em Wave Equation in conducting medium (σ ≠ 0)

The solution of the equation (14) should be a function of t and x and is of the form

Similarly, the solution of the equation (15) is of the form

Substituting eqn. (16) in egn. (14) 

For good conductors, we have σ >> ω ε. Therefore, μ ε ω2 can be neglected as compared to μ σ ω. Hence, from equation (18), we have

Taking square root on both sides

Taking the -ve value of y which gives the wave propagation in the +ve x-direction and substituting in the eqn. (16) we get

This is a progressive wave having amplitude equal to The amplitude of the wave goes on decreasing as the wave propagates deeper into the medium. Also the propagation constant k which is a constant depending upon the value of μ and σ.

Skin Depth (or) Penetration Depth

In conducting medium the amplitude of the electromagnetic wave decreases exponentially with distance of penetration of the wave. Suppose, the amplitude at a depth x is denoted by Eox , then

The decrease in the amplitude or the attenuation of the field vector is quantitatively expressed in terms of a quantity called skin depth.

It is defined as the distance inside the conductor from the surface of the conductor at which the amplitude of the field vector is reduced to l/e times its value at the surface.

Electromagnetic waves

According to Maxwell, an accelerated charge is a source of electromagnetic radiation.

• In an electromagnetic wave, electric and magnetic field vectors are at right angles to each other and both are at right angles to the direction of propagation.

They possess the wave character and propagate through free space without any material medium. These waves are transverse in nature.

• Fig. 2.6 shows the variation of electric field along Y direction and magnetic field along Z direction and wave propagation in +X direction.