Question

An electromagnetic wave of frequency$5GHz$, is travelling in a medium whose relative electric permittivity and relative magnetic permeability both are $2$. Its velocity in this medium is _______ $\times {10}^{7}m/s$.

Answer 1

Step 1. Given data:

$f=5GHz$, Frequency of electromagnetic wave

${\epsilon }_r=2$, Relative electric permittivity

${\mu }_r=2$, Relative magnetic permeability

Assumption:

$c=\frac{1}{\sqrt{{\mu }_o{\epsilon }_o}}=3\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$, Speed of an electromagnetic wave in vacuum.

${\mu }_0=$Absolute permeability and ${\epsilon }_0=$absolute permittivity

Step 2. Calculating velocity of electromagnetic wave:

The speed of the electromagnetic wave in the medium is $v=\frac{1}{\sqrt{{\mu }_o{\epsilon }_o{\mu }_r{\epsilon }_r}}$

$\Rightarrow$$v=\frac{1}{\sqrt{{\mu }_o{\epsilon }_o}\times \sqrt{{\mu }_r{\epsilon }_r}}$

$\Rightarrow$$v=\frac{c}{\sqrt{{\mu }_r{\epsilon }_r}}$ $\left[c=\frac{1}{\sqrt{{\mu }_o{\epsilon }_o}}\right]$

Putting all values in the above,

$\begin{array}{rcl}& \Rightarrow & v=\frac{3\times {10}^8}{\sqrt{2\times 2}}\\ & \Rightarrow & v=\frac{3\times {10}^8}{2}\\ & \Rightarrow & v=1.5\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.\\ \therefore v& =& 15\times {10}^7\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.\\ & & \end{array}$

The velocity of the wave in the medium is $\begin{array}{rcl}\mathbf{}\mathit{v}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{15}\mathbf{\times }{\mathbf{10}}^{\mathbf{7}}\mathbf{}\raisebox{1ex}{$\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{s}$}\right.\end{array}$.