Question

the incoming ray suffers total internal reflection (TIR) times as shown. The prism mhas reflective index $\mu$ and top angle $\theta$. The emerging ray propagates horizontally. The refrative index $\mu =\frac{cos\theta }{cosn\theta }$. Find n

Answer 1

From geometry, $\angle PBA=\theta$

As BN is perpendicular to PH i-e, $\angle PBN=90⟹\angle ABN=90-\theta$

At B TIR occurs, $⟹\angle NBC=\angle ABN=90-\theta$

Now $\angle FBC=\angle ABC-\angle ABF=2(90-\theta )-90=90-2\theta$

Also $\angle CDG=\angle LCD=\angle LCB=\angle FBC=90-2\theta$

$i=\angle CDG+\angle GDH-\angle MDH=90-2\theta +90-\theta -90⟹i=90-3\theta$

$r=90-\theta$

Using snell;s law at D, $\mu sini=1\times sinr$

$\mu sin(90-3\theta )=1\times sin(90-\theta )⟹\mu =\frac{cos\theta }{\cos 3\theta }$

Hence, $n=3$