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Electromagnetic Waves in Conducting medium

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1 Electromagnetic Waves in Conducting medium
Let us assume that medium is linear, homogeneous and isotropic and is characterized by permittivity , permeability , and conductivity , but not any charge or any current other than that determined by Omh’s law. Then Thus, Maxwell’s equations: and

2 Taking curl of 3rd equation:
Using 4th equation: From equation 1st:

3 Similarly we can find: These last two equations are wave equations governing electromagnetic waves in a homogeneous isotropic conducting medium. Let us assume that the fields vary as then solution of above equation may be expressed as: Substituting this value of E in the above equation, we get:

4 k is complex quantity here and can be written as:
or After comparing:

5 and By adding eq (1) and (2):

6 By subtracting eq (1) and (2):
Now, in terms of  and , the field vector E take the form: So, field amplitude are spatially attenuated (diminishes) due to the presence of the term so  is a measure of attenuation and is known as absorption coefficient.

7 Now, for good conductors:
and then, the medium may be classified as a conductor. and then, because: If; and then, the medium may be classified as a dielectric,

8 Skin depth or penetration depth
Wave in conducting medium gets attenuated. The distance it takes to reduce the amplitude by a factor of 1/e is called the skin depth. The term 1/ measures the depth at which electromagnetic waves entering a conductor is damped to 1/e =0.369. d = 1/ [= for good conductors] It is a measure of how far the wave penetrates into the conductor.

9 Retarded Potential Electromagnetic radiation (time varying electric fields especially) are produced by time varying electric charges. Since influence of charge (i.e. field) travels with a certain velocity so at a point, the effect of charge (i.e. potential at that point) is experienced after a certain time only. These kind of potentials are known as retarded potentials.

10 Suppose we have a varying charge (charge density (t)) and lets find the potential due to this charge at point P. A charge segment dV is at: The position vector of P is: Hence the position of P with respect to dV is:

11 Potential at P at time t, is due to charge segment dV at the time:
Hence the potential at P due to whole charge: Just to understand this form for V

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