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Appendix XXXIV
Solving the wave equation for plane wave
Maxwell’s four differential equations are:
In free space propogation: 𝝆( charge density) =0 and
Therefore the four differential equations become:
From these four equations we obtain:
But in free space both divergences are zero therefore the above 2 equations become:
In free space €=€ _{0} and μ= μ _{0}
That is free space permittivity = absolute permittivity .
And free space permeability = absolute permeability .
Also c ( velocity of light in free space ) =
Therefore the above two equations become:
We apply separation of variables to solve these two partial differential equations:
Assume E(r,t) = E1(r)E2(t)
H(r,t) = H1(r)H2(t)
In general by assuming the time dependent portion to be of harmonic nature:
That is E2(t) = E20Exp(jωt) and H2(t) = H20Exp(jωt)
The partial derivative
Hence the two equations become:
Now the partial differential equations have reduced to ordinary linear differential equations known as wave equations of 2 ^{nd} order and they can be solved applying operator theory.
We also assume that it is a plane wave travelling along z axis hence
Under such circumstances as we will see in Electro-Magnetic Field Theory, only x-component of E1 and y-component of H1 will remain. Hence
Applying Theory of Operator we get two roots in each case namely:
Two roots are =
= =Hence
So the complete solution for Electric Field is:
1 ^{st} Term is backward travelling wave and 2 ^{nd} term is forward travelling wave.
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