Question

Thickness of two transparent media A and B is $6:4$ . If time taken by light to pass from both media is the same, then the refractive index of B with respect to A is:
A) $1.33$
B) $1.4$
C) $1.5$
D) $1.75$

Answer 1

We know that, the refractive index of any medium is the ratio of speed of light in vacuum to the velocity of light in that medium which can be mathematically written as $\mu = \dfrac{c}{v}$ . According to the question, the ratio of thickness of two transparent media is given. So, we can convert the refractive index formula in terms of thickness as we know that velocity of any medium is the ratio of thickness to time. Also, the time is given the same for both of the media, so we can now equate the ratio of refractive index in terms of thickness to get the refractive index of B w.r.t A.

Formula used:
$\mu = \dfrac{c}{v}$
Where,
$\mu =$ Refractive index of the medium
$c =$ Speed of light in vacuum
$v =$ Velocity of light in the given medium
And $v = \dfrac{d}{t}$
Where,
$v =$ Velocity of light in the given medium
$d =$ Thickness of the medium
$t =$ Time taken by light to pass from the given medium

Complete step by step solution:
As we know, the refractive index is the ratio of speed of light in vacuum to the velocity of light in that medium.
So, for medium A
$\Rightarrow {\mu _A} = \dfrac{c}{{{v_A}}}$
For medium B,
$\Rightarrow {\mu _B} = \dfrac{c}{{{v_B}}}$
Now, calculating the refractive index of medium B w.r.t. A, we get
$\Rightarrow \dfrac{{{\mu _B}}}{{{\mu _A}}} = \dfrac{c}{{{v_B}}} \times \dfrac{{{v_A}}}{c} \\\ \Rightarrow \dfrac{{{\mu _B}}}{{{\mu _A}}} = \dfrac{{{v_A}}}{{{v_B}}}...\left( 1 \right) \\\$
Now, we know that velocity is the ratio of distance(thickness) to time.
According to the question, the time taken by light to pass through the medium is the same for both.
So, for medium A
${v_A} = \dfrac{{{d_A}}}{t}$
For medium B,
${v_B} = \dfrac{{{d_B}}}{t}$
Now, putting the values of velocity in equation (1)
$\Rightarrow \dfrac{{{\mu _B}}}{{{\mu _A}}} = \dfrac{{{d_A}}}{t} \times \dfrac{t}{{{d_B}}} \\\ \Rightarrow \dfrac{{{\mu _B}}}{{{\mu _A}}} = \dfrac{{{d_A}}}{{{d_B}}} \\\$
As given in question, the ratio of thickness of two transparent media A and B is $6:4$.
So, the refractive index of B w.r.t. B is:
$\Rightarrow \dfrac{{{\mu _B}}}{{{\mu _A}}} = \dfrac{{{d_A}}}{{{d_B}}} = \dfrac{6}{4} \\\ \Rightarrow {\mu _{BA}} = \dfrac{{{\mu _B}}}{{{\mu _A}}} = 1.5 \\\$
Hence, the correct answer is option C.

Note: The given question can be solved simply by using the direct formula of the refractive index and by replacing velocity components by the thickness and time. Then, we can input the given data and can get a refractive index. In these types of questions, we just need to reduce the formula in the terms whose values are provided in the question.