Question

A transparent cube of $15cm$ edge contains a small air bubble. Its apparent depth when viewed through one face is $6cm$ and when viewed through the opposite face is $4cm$. Then the refractive index of the material of the cube is:
A. $2.0$
B. $2.5$
C. $1.6$
D. $1.5$

Answer 1

In order to solve this question you have to know the relationship between the real depth, apparent depth, and the refractive index of the material. Also, you have to know the concept of refraction of light in order to understand the real and apparent depth.

Formula used:
The Refractive index of the material is given by,

$\mu = \dfrac{{{D_{real}}}}{{{D_{apparent}}}}$

Where $\mu$ is the refractive index of the material

${D_{real}}$ is the real depth of the object
${D_{apparent}}$ is the apparent depth of the object

Complete step by step solution:
Here, in the question, it is given that the actual depth is $15cm$
And also given that the apparent depth, when viewed through one face, is $6cm$
So, let us consider the bubble be at a distance $xcm$ when viewed through one face
Hence, according to the formula of the refractive index of the material, we have

$\mu = \dfrac{{{D_{real}}}}{{{D_{apparent}}}}$

On putting the required values, we have

$\Rightarrow \mu = \dfrac{x}{6}$ ……….(i)

Now, when viewed through the opposite face we have the real depth as $(15 - x)cm$ and the apparent depth is given in the question that is $4cm$.
On putting in the formula we have,

$\Rightarrow \mu = \dfrac{{15 - x}}{4}$ ……..(ii)

Now, from equation (i) and (ii), we get

$\Rightarrow \dfrac{x}{6} = \dfrac{{15 - x}}{4}$

We have to solve the above equation for finding the value of $x$

$\Rightarrow 4x = 90 - 6x$

On further solving, we get

$\Rightarrow x = 9$

Now, substitute the value in the equation (i), we get

$\Rightarrow \mu = \dfrac{9}{6}$

On further solving, we have

$\mu = 1.5$
Thus, the refractive index of the material of the cube is $1.5$

Therefore, the correct option is (D).

Note: Remember that the real depth is the actual distance of an object which is under the surface and can be measured by submerging a perfect ruler while apparent depth is the distance of an object in a denser medium when seen from the rarer medium. Also, remember that the value of apparent depth is smaller than the real depth if the refractive index of the material is greater than the medium in which the observer is present. Also, the converse is also true.