Question 33: EM waves

Question

The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant $k=3$ and permeability of $\mu=2 \mu_0$, is ( $\mu_0=$ permeability of vacuum)

Accepted Answer

The speed of light (or any electromagnetic wave) in a vacuum is given by the formula :
$$ c=\frac{1}{\sqrt{\mu_0 \epsilon_0}} $$
Where :
1. $\mu_0$ is the permeability of free space.
2. $\epsilon_0$ is the permittivity of free space.
The speed of light in a specific medium is determined by that medium's permeability ( $\mu$ ) and permittivity ( $\epsilon$ ):
$$ v=\frac{1}{\sqrt{\mu \epsilon}} $$
For the given medium :
Permeability : $\mu=2 \mu_0$
Dielectric Constant $\mathrm{k}=3$. The dielectric constant (relative permittivity $\epsilon_{\mathrm{r}}$ ) is defined as $\epsilon_{\mathrm{r}}=\frac{\epsilon}{\epsilon_0}$. Therefore, $\epsilon=\mathrm{k} \epsilon_0=3 \epsilon_0$.
We need to find the ratio $c: v$, which is the same as the refractive index $n$ of the medium.
$$ \frac{c}{v}=\frac{1 / \sqrt{\mu_0 \epsilon_0}}{1 / \sqrt{\mu \epsilon}} $$
$$\Rightarrow $$ $$ \frac{c}{v}=\sqrt{\frac{\mu \epsilon}{\mu_0 \epsilon_0}}=\sqrt{\frac{\left(2 \mu_0\right)\left(3 \epsilon_0\right)}{\mu_0 \epsilon_0}} $$
For the medium $\mu=2 \mu_0$ and $\epsilon=3 \epsilon_0$ :
$$ \frac{c}{v}=\sqrt{\frac{\left(2 \mu_0\right)\left(3 \epsilon_0\right)}{\mu_0 \epsilon_0}}=\frac{\sqrt{6}}{1} $$
The ratio of the speed in vacuum to the speed in the medium is $\sqrt{6}: 1$.
Hence, the correct option is (C).

Answer

$6: 1$

Answer

$3: 2$

Answer

$\sqrt{6}: 1$

Answer

$36: 1$

EM waves

The ratio of speeds of electromagnetic waves in vacuum and a medium, having dielectric constant k=3 and permeability of \mu=2 \mu_0, is ( \mu_0= permeability of vacuum)

Choose the correct answer: