Question

An equilateral prism deviates a ray through ${23}^{\circ }$ for two angles of incidence differing by ${23}^{\circ }$. Find $\mu$ of the prism?

• A. $\frac{\sqrt{33}}{5}$
• B. $\frac{\sqrt{43}}{3}$
• C. $\frac{\sqrt{43}}{5}$
• D. $\frac{\sqrt{62}}{2}$

Answer 1

The correct option is C $\frac{\sqrt{43}}{5}$

Given that
$i-e={23}^{\circ }......\left(1\right)$
Deviation in this case is
$23=i+e-{60}^{\circ }\Rightarrow i+e={83}^{\circ }....\left(2\right)$
From equation (1) and (2),
$e={30}^{\circ }\&i={53}^{\circ }$


Applying snells law, at refracting surfaces of prism
$1.\sin {53}^{\circ }=\mu \sin r.....\left(3\right)$
$\Rightarrow \mu \sin r=\mathrm{0.8....}\left(4\right)$
$\mu \sin (60-r)=\sin \mathrm{30....}\left(5\right)$

$\mu \sin \left(60\right)\cos r-\mu \cos 60\sin r=\sin \mathrm{30....}\left(4\right)$
$\mu \cos r\sqrt{3}-\mu \sin r=1$
$\mu \cos r=1.8/\sqrt{3}...\left(6\right)$
On squaring and adding $4\&6,$ we get
$\mu =\sqrt{\frac{64}{100}+\frac{324}{300}}$
$\mu =\frac{\sqrt{43}}{5}$