Question

A ray of light AO in vacuum is incident on a glass slab at angle 60° and refracted at angle 30° along OB as shown in the figure.
The optical path length of light ray from A to B is

• A. $\frac{2\sqrt{3}}{a}+2b$
• B. $2a+\frac{2b}{3}$
• C. 2a + 2b
• D. $2a+\frac{2b}{\sqrt{3}}$

Answer 1

The correct option is C 2a + 2b

From the figure,


$\cos 60{}^{\circ }=\frac{a}{AO}$

$\Rightarrow AO=\frac{a}{\cos 60{}^{\circ }}=2a...\left(i\right)$

$and\cos 30{}^{\circ }=\frac{b}{BO}$

$orBO=\frac{b}{\cos 30{}^{\circ }}=\frac{2b}{\sqrt{3}}....\left(ii\right)$
Optical path length of light ray
= AO + $\mu$ (BO) ... (iii)

Here, $\mu$ can be determined using Snell's law, i.e.

$\mu =\frac{\sin 60{}^{\circ }}{\sin 30{}^{\circ }}=\frac{\sqrt{3}/2}{1/2}=\sqrt{3}....\left(iv\right)$
Substituting the values from Eqs. (i), (ii) and (iv)

$\therefore Opticalpath=2a+(\sqrt{3}\times \frac{2}{\sqrt{3}}b)$
= 2a + 2b