The problem of determining the surface current density induced on an antenna surface placed in a nonlinear inhomogeneous and anisotropic medium taking gravitational effects into account

Statement of the problem: Let e^(w, r, F) denote the field dependent permittivitypermeability tensor. Here,

Eup = Ap n —

is the covariant em field tensor with A/( as the covariant em four potential, the dependence of on the em Held tensor F = ((FMI/)) shows that the medium is nonlinear and its dependence on the space-time coordinates x and the fact that this tensor is generally non-diagonal shows that the medium is nonlinear, inhomogeneous and anisotropic. The Maxwell field equations in such a medium are

(^(W,r,F)F“^)(IZ = ^(W,æ)

yields the nonlinear wave equation in such a medium taking space-time curvature into account. We assume the Lorentz gauge conditions

= 0

Here,

A>‘ = g^Av, = g^g^Faa

We are here assuming that the gravitational field metric tensor is time independent and hence we are operating completely in the temporal frequency domain. Thus where-ever do occurs in the above system, we replace it by the multiplication operator jw.

= AM(w,r),

■^Or j^Ar AOr,

9nv = P^(r)

where r = (x,y, z). The above Maxwell equations can be expressed as

>$(u>,r,F)F^y^ + (e^F^^),m = 0

To simplify further, we must make use of the gauge condition:

{Ayy^gg^Y^ = 0

gives

jcjgOv y/^gAv + {Avgmv y/^gYm = 0

If we use the synchronous reference system, then go™ = 0 and goo = 1- Then the metric can be expressed as

dr2 = dt2 — 'Yrs(r)dxr dxs

This equation implies that gOm = 0 and g°° = 1. So we get for the gauge condition,

juy/^Ao - (AsymsyYm = 0

where ((qrs)) = ((7rS))-1-

Reference: L.D.Landau and E.M.Lifshitz, ’’The classical theory of fields”, Butterworth and Heinemann.

Acknowledgements: I am grateful to Prof.Malay Ranjan Tripathi for suggesting me to write this article for a workshop conducted by him.

 
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