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# Complex Refractive Index and Power Reflection Coefficient

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Show that the complex refractive index of a conducting medium can be expressed as n = λ0(1+i)/2 πδ and n = λ0(1-i)/2πδ where δ is the skin depth. Hence find the power reflection coefficient for an EM wave incident from free space on a good non-magnetic conductor at normal incidence. Show that the energy lost per unit area in the conductor is
H0^2/2σδ, where H0 is the amplitude of the magnetic field in the metal just beneath the surface and σ is the conductivity.

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#### Solution Preview

Please refer to the attached document for the solution.

Show that the complex refractive index of a conducting medium can be expressed as n = λ0(1+i)/2πδ and n = λ0(1-i)/2πδ where δ is the skin depth.

Consider the following situation in your problem:

Here is the explanation of the above figure:

We are dealing with a semi-infinite conducting permeable medium, where μ is the permeability of the material and ε is the dielectric function of the material. δ is the skin depth of the conducting material.

Here comes the problem..... Let's apply a spatially constant, but time varying magnetic field in the x-direction in the form H(x,t) = Ho Cos(wt). Our problem is to find a solution for the steady state with the proper boundary conditions for z > 0 and z -> infinity.

At this point you need to know the diffusion equation in the scalar vector potential form:

You can use this diffusion equation also for the electric and magnetic fields.

Note.- You can derive the above formula from Maxwell Equations tighter with the idea of expressing the magnetic field and electric field in terms of a vector potential A.

Try to ...

#### Solution Summary

This response provides guidelines on showing the energy lost per unit area in a conducting medium.

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