Question

A Narrow cylindrical beam of light falls on a spherical air bubble, located in a liquid so that the axis of the beam passes through the center of the bubble. Determine the refractive index of the liquid, if it is known that the area of the beam at the exit from the bubble is 4 times the cross-sectional area at the entrance.
A. $1.5$
B. $\dfrac{4}{3}$
C. $2$
D. $1.75$

Answer 1

Use the law of refraction in a curve surface to find the refractive index of the liquid when the object is placed at infinite. The law of refraction for an object placed in denser medium and when beam is diverging is given by, $- \dfrac{{{n_1}}}{u} + \dfrac{{{n_2}}}{v} = \dfrac{{{n_1} - {n_2}}}{R}$.

Complete step by step answer:
We know that, the law of refraction for an object placed in denser medium and when beam is diverging can be written as, $- \dfrac{{{n_2}}}{u} + \dfrac{{{n_1}}}{v} = \dfrac{{{n_1} - {n_2}}}{R}$. Here, ${n_2}$ is the refractive index of the denser medium, ${n_1}$ is the refractive index of the rarer medium, $R$ is the curvature of the surface, $u$ is the object distance and , $v$ is the image distance.

Now, a parallel beam of light falls on the spherical bubble. So, the object is at infinity and the light is passing from denser medium (liquid ) to the rarer medium (air inside the bubble).Hence, here we have, object distance $u = \infty$. ${n_1}$ is the refractive index of the liquid, ${n_2} = 1$ is the refractive index of air inside the bubble, $v = 2R$, since the beam of light at the other end is four times the entering beam so the image must be forming at the other end of the bubble. So, putting the values we get,
$- \dfrac{1}{\infty } + \dfrac{{{n_1}}}{{2R}} = \dfrac{{{n_1} - 1}}{R}$
On simplifying we get,
$0 + \dfrac{{{n_1}}}{2} = {n_1} - 1$
$\Rightarrow 2{n_1} - 2 = {n_1}$
$\therefore {n_1} = 2$

Hence, the correct answer is option C.

Note: The area of the image or object is not taken in consideration for calculation. Since, the only necessary conditions here are the distance of the object from the surface and the distance of the image from the surface.As, the beam of light is coming in parallel so the object is taken to be at infinite means at a very large distance.