Question

The ratio of thickness of plates of two transparent medium A and B is 6:4. If it takes equal time in passing through them, then refractive index of B with respect to A will be
(A) 1.4
(B) 1.5
(C) 1.75
(D) 1.33

Answer 1

Hint: Refractive index of a material depends upon the speed of light in that material. It is the ratio of speed of light in vacuum to speed of light in the medium. More the speed, less the time it requires for the light to travel some given distance in a medium.
Formula used:
The relation of refractive index and the velocity of light in that medium is:
$\eta = \dfrac{c}{v}$

Complete answer:
We are given that the ratio of the thickness of the two plates is 6:4.
Let us assume that the first plate has a thickness 6a and the second has a thickness of 4a. Let the time required to pass through them be t. Now, we are also given that the time required to pass through them is the same.
Now we know that distance by time is velocity. Therefore, for the case of plate 1, the velocity is:
$\dfrac{6a}{t} = 6 v$ .
For the case of plate 2, the velocity is:
$\dfrac{4a}{t} = 4 v$.
Where a replacement of a/t with another variable v for the velocity has been made.
Now, looking at the refractive index expression, we find that:
$\eta_A = \dfrac{c}{6v}$
and,
$\eta_B = \dfrac{c}{4v}$ .
Dividing the two expressions we get:
$\dfrac{\eta_A}{\eta_B} = \dfrac{4}{6} = \dfrac{2}{3}$
Now, the refractive index of B with respect to A can be written as follows:
$ \eta_{BA} = \dfrac{\eta_B}{\eta_A} = \dfrac{3}{2} = 1.5$

Therefore, the correct answer is option (B).

Note:
The definition for the refractive index of a medium with respect to another can be a little tricky to remember and one can easily obtain the inverse ratio if that part goes wrong. The notation goes as $ \eta_{12}$ where we find the refractive index of medium 1 with respect to medium 2 which happens to be the ratio of $n_1$ over $n_2$.