Question

If $c$ is the speed of electromagnetic waves in vacuum then $v$, its speed in a medium of dielectric constant $K$ and relative permeability ${\mu }_r$ is

• A. $v=\frac{1}{\sqrt{{\mu }_{r}k}}$
• B. $v=c\sqrt{{\mu }_{r}k}$
• C. $v=\frac{c}{\sqrt{{\mu }_{r}k}}$
• D. $v=k\sqrt{{\mu }_{r}c}$

Answer 1

The correct option is C

$v=\frac{c}{\sqrt{{\mu }_{r}k}}$

Step 1: Given data

$c\to$speed of electromagnetic waves in vacuum

$v\to$speed of electromagnetic waves in another medium

$k\to$dielectric constant

${\mu }_r\to$relative permeability

Step 2: Formula used

Speed of electromagnetic wave in vacuum, $c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}\to \left(1\right)$

Speed of electromagnetic wave in another medium, $v=\frac{1}{\sqrt{\mu \epsilon }}\to \left(2\right)$

Step 3: Calculation

Dividing $\left(1\right)$ by $\left(2\right)$, we get

$$\begin{gathered} \frac{c}{v}=\frac{\sqrt{\mu \epsilon }}{\sqrt{{\mu }_0{\epsilon }_0}} \\ \Rightarrow \frac{c}{v}=\sqrt{\frac{\mu }{{\mu }_0}}\times \sqrt{\frac{\epsilon }{{\epsilon }_0}}\left(Weknow,\sqrt{\frac{\mu }{{\mu }_0}}={\mu }_{r}and\sqrt{\frac{\epsilon }{{\epsilon }_0}}=k\right) \\ \Rightarrow \frac{c}{v}=\sqrt{{\mu }_{r}k} \\ \Rightarrow v=\frac{c}{\sqrt{{\mu }_{r}k}} \end{gathered}$$

So, we get the relation between $c,v,k$ and ${\mu }_r$ is$v=\frac{c}{\sqrt{{\mu }_{r}k}}$

Hence, option C is correct