Question

A beam of light on a glass hemisphere as shown. If the radius of curvature of the hemisphere is very large in comparison to d, then the diameter of the beam at the base of the hemisphere will be:

• A. $\frac{3}{4}d$
• B. $d$
• C. $\frac{d}{3}$
• D. $\frac{2}{3}d$

Answer 1

The correct option is D $\frac{2}{3}d$

Given, $u=\infty$

General equation for refraction through a curved surface is given as

$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$

$\Rightarrow \frac{3/2}{v}-\frac{1}{\infty }=\frac{3/2-1}{R}$

$\Rightarrow \frac{3}{2v}=\frac{1}{2R}$

$\therefore v=3R$

From the figure we can see that $\Delta ABC$ and $\Delta ADE$ are similar triangles.

$\therefore \frac{BC}{ED}=\frac{AG}{AF}$

$\Rightarrow \frac{x}{d}=\frac{2R}{3R}$

$\Rightarrow x=\frac{2}{3}d$

Hence the correct answer is option (D).