Question

An air bubble in a glass with refractive index $1.5$ (near normal incidence) i.e. $5\,cm$ deep when viewed from one surface and $3\,cm$ deep when viewed from the opposite face. The thickness is:
(A) $16$
(B) $8$
(C) $10$
(D) $12$

Answer 1

Hint
First by using the given refractive index and the apparent depth value, the real depth is determined. Then by using the second apparent depth value, the second real depth can be determined, by adding the two real depth values, the thickness can be determined.
The refractive index of the air bubble is given by,
$\Rightarrow \mu = \dfrac{{{d_1}}}{{{d_2}}}$
Where, $\mu $ is the refractive index of the air bubble, ${d_1}$ is the real depth and ${d_2}$ is the apparent depth.

Complete step by step answer
Given that, The refractive index of the air bubble is, $\mu = 1.5$.
The first apparent depth value is, ${d_2} = 5\,cm$.
The second apparent depth value is, ${d_2} = 3\,cm$.
Now, The refractive index of the air bubble is given by,
$\Rightarrow \mu = \dfrac{{{d_1}}}{{{d_2}}}\,..................\left( 1 \right)$
By substituting the refractive index of the air bubble and the first apparent depth in the above equation (1), then
$\Rightarrow 1.5 = \dfrac{{{d_1}}}{5}$
By rearranging the terms in the above equation, then
$\Rightarrow {d_1} = 1.5 \times 5$
By multiplying the terms in the above equation, then
$\Rightarrow {d_1} = 7.5\,cm$
Now, substituting the refractive index of the air bubble and the second apparent depth in the above equation (1), then
$\Rightarrow 1.5 = \dfrac{{{d_1}}}{3}$
By rearranging the terms in the above equation, then
$\Rightarrow {d_1} = 1.5 \times 3$
By multiplying the terms in the above equation, then
$\Rightarrow {d_1} = 4.5\,cm$
The thickness of the air bubble is, $ \Rightarrow 7.5\,cm + 4.5\,cm$.
Then the thickness is $12\,cm$.
Hence, the option (D) is the correct answer.

Note
The refractive index is directly proportional to the real depth and the refractive index is inversely proportional to the apparent depth. As the refractive index increases when the real depth increases and the apparent depth decreases. As the refractive index decreases when the real depth decreases and the apparent depth increases.