Question

The velocity of light in diamonds is $ 121000km/s $ . What is the refractive index?
(A) $ \;1.47 $
(B) $ \;2.47 $
(C) $ \;3.52 $
(D) $ \;3.86 $

Answer 1

Hint: The velocity of light in different medium varies with respect to the refractive index. From the given data in the question, we can write the relation between the refractive index and the velocity of the light in the air in the diamond. Then find the refractive index.

Formula used
 $ \mu = \dfrac{{{v_o}}}{{{v_m}}} $

Complete Step-by-step solution
The refractive index of a medium can be mathematically in expressions of velocity as,
 $ \mu = \dfrac{{{v_o}}}{{{v_m}}} $
where,
 $ \mu $ is the refractive index of the medium
 $ {v_o} $ is the velocity of light in air
 $ {v_m} $ is the velocity of the light in the medium
They’ve given the question that the velocity of light in a diamond is $ 121000km/s $ . First, we have to change $ km/s $ to $ m/s $ . For this, we have to multiply it with $ \;1000 $
 $ \Rightarrow 121000 \times 1000 $
 $ \Rightarrow 121 \times {10^6}m/s $
Substituting these values in the above formula we have
 $ {\mu _{diamond}} = \dfrac{{{v_o}}}{{{v_{diamond}}}} $
Since we know that the speed of light in air is $ 3 \times {10^8}m/s $ .
Substitute all the given data in the above equation we get,
 $ \Rightarrow {\mu _{diamond}} = \dfrac{{3 \times {{10}^8}}}{{121 \times {{10}^6}}} $
 $ \Rightarrow {\mu _{diamond}} = \dfrac{{3 \times {{10}^8}}}{{1.21 \times {{10}^8}}} $
On further solving the equation we get,
 $ \Rightarrow {\mu _{diamond}} = \dfrac{3}{{1.21}} $
 $ \Rightarrow {\mu _{diamond}} = 2.47 $
Therefore, the refractive index of light in the diamond is $ \;2.47 $ .
Hence, the correct answer is option (B).

Additional information
The refractive index of a medium is a dimensionless number that explains how fast light can travel through the medium. When the light enters a substance with a higher refractive index, then the angle of refraction will be smaller than the angle of incidence. Therefore the light gets refracted towards the normal of the surface. The higher the refractive index, the closer the ray will be to the normal.

Note
The refractive index does not have any units as it is the ratio of two similar quantities. Yet we know the standard speed of light, for air or vacuum it is always better to find it out from the given data when it is not stated. This will make sure the accuracy of the calculations.