Question

Given that the EM wave has a form of Ex=Eo cos(wt-kz+∅) and it can also be written as Ex=Eo[k(vt-z)+∅] where v=w/k is the velocity.

(a)show that the wave satisfy Maxwell equation

(b)show that v=cuberoot of permeability note,permetivity note permetivity

Answer 1

$E_x=E_ocos(\omega t-kx+\phi_o)\\\dfrac{d^2E_x }{dx^2}=0\ \ \ \ \dfrac{d^2E_x }{dy^2}=0\ \ \ \dfrac{d^2E_x }{dz^2}=-k^2E_ocos(\omega t-kz+\phi_o)......(i)$

$\dfrac{d^2E_x }{dt^2}=-\omega^2E_ocos(\omega t-ky+\phi_o)......(ii)$

Substituting (i) and (ii) in wave equation we get,

$-k^2E_ocos(\omega t-kz+\phi_o) + \epsilon_o\epsilon_r\mu_o+\omega^2E_ocos(\omega t-ky+\phi_o)=0\\ \Rightarrow\dfrac{\omega^2}{k^2}=\dfrac{1}{ \epsilon_o\epsilon_r\mu_o}\\ \Rightarrow \dfrac{\omega}{k}=( \epsilon_o\epsilon_r\mu_o)^{-\frac{1}{2}}\\ \Rightarrow v=(\sqrt{ \epsilon_o\epsilon_r\mu_o})^{-1}$