A plane wave of angular frequency and wave number propagates through a transparent…

PHYSICS · Multi-part · CBSE · Physics · Electromagnetic Waves · Electromagnetic Waves

Problem

A plane wave of angular frequency

\omega

and wave number

k

propagates through a transparent non-conducting medium having permeability

\mu

and permittivity

\varepsilon

. The electric field is

E(x,t)=E_0\sin(kx-\omega t).

Solve the following parts. (i) Show that the given field satisfies the wave equation

\nabla^{2}E=\mu\varepsilon\,\dfrac{\partial^{2}E}{\partial t^{2}}

only if

\dfrac{\omega}{k}=\dfrac{1}{\sqrt{\mu\varepsilon}}

. Provide full working. (ii) Sea water has relative permittivity

\varepsilon_r=80

and relative permeability

\mu_r\approx1

. Calculate the speed of the electromagnetic wave in sea water. (iii) If

k=1.5\times10^{7}\,\text{m}^{-1}

for a wave in sea water, compute the required value of

\omega

so that the field in part (i) represents a valid electromagnetic wave in that medium.