Question

A light ray is incident normally on the face AB of a right-angled prism ABC (μ = 1.50) as shown in figure. What is the largest angle ϕ for which the light ray is totally reflected at the surface AC?
Figure

Answer 1

Given,
Refractive index (μ) of prism = 1.50
Let us take θ_c as the critical angle for the glass.

So, According to Snell's law,
$\frac{\sin {\mathrm{\theta }}_c}{\sin 90°}=\frac{1}{\mu }$
$\Rightarrow \sin \mathrm{\theta }c=\frac{1}{1.5}=\frac{2}{3}$
$\Rightarrow {\theta }_{\mathrm{c}}={\sin }^{-1}\frac{2}{3}$

Condition for total internal reflection: 90° − $\varphi$> θ_c
⇒ $\varphi$ < 90° $-$ θ_c
$\Rightarrow \varphi <90°-{\sin }^{-1}\left(\frac{2}{3}\right)$
$\Rightarrow \varphi <{\cos }^{-1}\left(\frac{2}{3}\right)$
Hence, the largest angle for which light is totally reflected at the surface AC is ${\cos }^{-1}\left(\frac{2}{3}\right)$.