In a medium, the speed of light wave decreases to 0.2 times its s… — Solution | JEE Physics

NVAEasyJEE 2023Characteristics of EM Waves

In a medium, the speed of light wave decreases to $0.2$ times its speed in free space. The ratio of relative permittivity to the refractive index of the medium is $x : 1$ . The value of $x$ is _____. (Given speed of light in free space $c = 3 \times 10^8 \, \text{m/s}$ and for the given medium $\mu_r = 1$ )

Answer

Correct answer:5

Step-by-step solution

Standard Method

Given: Speed of light in the medium is $0.2c$ and $\mu_r = 1$ .

Find: The value of $x$ in the ratio $\epsilon_r : n = x : 1$ .

The refractive index is related to the speed of light by

n = \frac{c}{v}

Since

v = 0.2c

we get

n = \frac{c}{0.2c} = 5

Also, for a medium,

n = \sqrt{\mu_r \epsilon_r}

Given $\mu_r = 1$ , so

n = \sqrt{\epsilon_r}

Hence,

\epsilon_r = n^2 = 25

Now the required ratio is

\frac{\epsilon_r}{n} = \frac{25}{5} = 5

Therefore, the value of $x$ is $5$ .

Using extracted solution relation

Given: The wave speed in the medium is $0.2$ times the speed in free space.

Find: The ratio of relative permittivity to refractive index.

From the extracted working,

n = \frac{c}{v}

n = \frac{c}{0.2c} = 5

Using

n = \sqrt{\epsilon_r \mu_r}

and $\mu_r = 1$ , we have

5 = \sqrt{\epsilon_r}

Therefore,

\epsilon_r = 25

So,

\epsilon_r : n = 25 : 5 = 5 : 1

Hence, $x = 5$ .

Common mistakes

Using $n = \frac{v}{c}$ instead of $n = \frac{c}{v}$ . This reverses the refractive index and gives a value less than $1$ , which is incorrect here. Always use speed in free space divided by speed in the medium.
Forgetting that $\mu_r = 1$ . If you use $n = \sqrt{\epsilon_r \mu_r}$ without substituting $\mu_r = 1$ , you may leave an unnecessary extra variable. First simplify to $n = \sqrt{\epsilon_r}$ .
Taking the required ratio as $\frac{n}{\epsilon_r}$ instead of $\frac{\epsilon_r}{n}$ . The question asks for relative permittivity to refractive index, so the order matters. Read the ratio carefully before calculating.

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