Question

The velocity of light in glass whose refractive index w.r.t air is $1.5$ is $2 \times {10^8}m{s^{ - 1}}$. In a certain liquid, the velocity of light is found to be $2.5 \times {10^8}m{s^{ - 1}}$. The refractive index of the liquid w.r.t air is
A. $0.8$
B. $0.9$
C. $1.2$
D. $1.33$

Answer 1

The velocity of light in different medium varies with respect to the refractive index. Using the relationship between the speed of the light and refractive index one can know the speed of the light specific to the medium. From the given data in the question we can write the relation between refractive index and velocity of the light in the glass and the liquid. Then find the refractive index of the liquid.
Formula used:
$\mu = \dfrac{{{v_o}}}{{{v_M}}}$

Complete answer:
The refractive index of a medium can be mathematically in terms 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 glass of refractive index $1.5$ is $2 \times {10^8}m{s^{ - 1}}$ . Substituting these values in the above formula we have
\eqalign{ & {\mu _{glass}} = \dfrac{{{v_o}}}{{{v_{glass}}}} \cr & \Rightarrow {v_o} = {\mu _{glass}} \times {v_{glass}} \cr & \Rightarrow {v_o} = 1.5 \times 2 \times {10^8}m{s^{ - 1}} \cr & \Rightarrow {v_o} = 3 \times {10^8}m{s^{ - 1}} \cr}
Now that we have the velocity of light in air, we can find the velocity of light in the liquid to be
\eqalign{ & {\mu _{liquid}} = \dfrac{{{v_o}}}{{{v_{liquid}}}} \cr & \Rightarrow {\mu _{liquid}} = \dfrac{{3 \times {{10}^8}m{s^{ - 1}}}}{{2.5 \times {{10}^8}m{s^{ - 1}}}} = 1.2 \cr & \therefore {\mu _{liquid}} = 1.2 \cr}
Therefore, the refractive index of the liquid is 1.2.

So, the correct answer is “Option C”.

Note:
The refractive index does not have any units as it is the ratio of two similar quantities. Though 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 specified. This will ensure the accuracy in the calculations.