# Propagation of plane EM wave in conducting medium (or) lossy dielectrics

A lossy dielectric medium is one which an EM wave as it propagates losses power owing to imperfect dielectric,that is a lossy dielectric is an imperfect conductor that is a partially conducting medium $(\sigma&space;\neq&space;0)$ .

where as a lossless dielectric is a  $(\sigma&space;=0)$ perfect dielectric,then wave equations for conductors are also holds good here

i.e, $\bigtriangledown&space;^{2}\overrightarrow{E}=\mu&space;\sigma&space;\frac{\partial&space;\overrightarrow{E}}{\partial&space;t}&space;+\mu&space;\epsilon&space;\frac{\partial&space;^{2}\overrightarrow{E}}{\partial&space;t^{2}}$

$\frac{\partial&space;}{\partial&space;t}=&space;jw$

then $\bigtriangledown&space;^{2}\overrightarrow{E}=&space;j\omega&space;\mu&space;\sigma&space;\overrightarrow{E}+\mu&space;\epsilon(j\omega&space;)&space;^{2}\overrightarrow{E}$

$\bigtriangledown&space;^{2}\overrightarrow{E}=&space;(\sigma&space;+&space;j&space;\omega&space;\epsilon&space;)j\omega&space;\mu&space;\overrightarrow{E}$

$\bigtriangledown&space;^{2}\overrightarrow{E}=&space;\gamma&space;^{2}&space;\overrightarrow{E}$

$\bigtriangledown&space;^{2}\overrightarrow{E}-&space;\gamma&space;^{2}&space;\overrightarrow{E}=0---------Equation&space;(1)$

Equation (1) is called helm holtz equation and $\gamma$ is  called propagation constant.

$\gamma&space;^{2}&space;=j\omega&space;\mu&space;(\sigma&space;+j\omega&space;\epsilon&space;)$

$\gamma&space;^{2}=&space;j\omega&space;\mu&space;\sigma&space;-\omega&space;^{2}\mu&space;\epsilon$

Since $\gamma$ is a complex quantity it can be expressed as $\gamma&space;=&space;\alpha&space;+j\beta$

$\alpha$– is attenuation constant measured in Nepers/meter.

$\beta$-is phase constant measured in radians/meter.

$(\alpha&space;+j\beta&space;)&space;^{2}=&space;j\omega&space;\mu&space;\sigma&space;-\omega&space;^{2}\mu&space;\epsilon$

$\alpha^{2}&space;+2j\alpha&space;\beta-\beta&space;^{2}&space;=&space;j\omega&space;\mu&space;\sigma&space;-\omega&space;^{2}\mu&space;\epsilon$

by equating real and imaginary parts separately $\alpha&space;^{2}-\beta&space;^{2}=&space;-\omega&space;^{2}\mu&space;\epsilon------Equation(2)$

and $2\alpha&space;\beta&space;=\omega&space;\mu&space;\sigma$

$\alpha&space;=\frac{\omega&space;\mu&space;\sigma}{2\beta&space;}$

by substituting  $\alpha$ value in the equation (2)   $\frac{\omega&space;^{2}\mu&space;^{2}\sigma&space;^{2}}{4\beta&space;^{2}}-\beta&space;^{2}=-\omega&space;^{2}\mu&space;\epsilon$

${\omega&space;^{2}\mu&space;^{2}\sigma&space;^{2}}-4\beta&space;^{4}=-4\omega&space;^{2}\beta&space;^{2}\mu&space;\epsilon$

$4\beta&space;^{4}-4\omega&space;^{2}\beta&space;^{2}\mu&space;\epsilon&space;-{\omega&space;^{2}\mu&space;^{2}\sigma&space;^{2}}=0$

let $\beta&space;^{2}=t$

$4t^{2}-4\omega&space;^{2}t\mu&space;\epsilon&space;-{\omega&space;^{2}\mu&space;^{2}\sigma&space;^{2}}=0$

$t^{2}-\omega&space;^{2}t\mu&space;\epsilon&space;-\frac{\omega&space;^{2}\mu&space;^{2}\sigma&space;^{2}}{4}=0$

the roots of the above quadratic expression are

$t=\frac{\omega&space;^{2}\mu&space;\epsilon&space;\pm&space;\sqrt{\omega&space;^{4}\mu&space;^{2}\epsilon&space;^{2}-4(-\frac{\omega&space;^{2}\mu&space;^{2}\sigma&space;^{2}}{4})}}{2}$

$t=\frac{\omega&space;^{2}\mu&space;\epsilon&space;\pm&space;\sqrt{\omega&space;^{4}\mu&space;^{2}\epsilon&space;^{2}(1+\frac{\sigma&space;}{\omega&space;\epsilon&space;})^{2}}}{2}$

$\beta&space;^{2}=\frac{\omega&space;^{2}\mu&space;\epsilon&space;\pm&space;\sqrt{\omega&space;^{4}\mu&space;^{2}\epsilon&space;^{2}(1+\frac{\sigma&space;}{\omega&space;\epsilon&space;})^{2}}}{2}$

$\beta&space;=\sqrt{\frac{\omega&space;^{2}\mu&space;\epsilon&space;\pm&space;\sqrt{\omega&space;^{4}\mu&space;^{2}\epsilon&space;^{2}(1+\frac{\sigma&space;}{\omega&space;\epsilon&space;})^{2}}}{2}}$

$\beta&space;=\sqrt{\frac{\omega&space;^{2}\mu&space;\epsilon&space;(1+&space;\sqrt{(1+\frac{\sigma&space;}{\omega&space;\epsilon&space;})^{2}})}{2}}$

similarly,

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