Question

A ray of light is incident at an angle of incidence ${45}^{\circ }$ on the face of a prism having angle of the prism ${30}^{\circ }$. The refracted ray does not change its direction as it crosses the other face and emerges out of the prism. Find:
(a) the refractive index of the material of the prism and
(b) the angle of deviation.

Answer 1

Given:
The angle of incidence, $\angle i={45}^{\circ }$
The angle of prism, $\angle A={30}^{\circ }$
The angle of emergence, $\angle r_2=0^{\circ }$



(a) We know that, $A=r_1+r_2$
$\therefore {30}^{\circ }=r_1+0^{\circ }$
$⟹r_1={30}^{\circ }$
So, the angle of refraction on the first face is ${30}^{\circ }$.
Using Snell's law,
${\mu }_{21}=\frac{\sin i_1}{\sin r_1}=\frac{\sin {45}^{\circ }}{\sin {30}^{\circ }}$
$⟹{\mu }_{21}=\sqrt{2}$ (refractive index of prism)

(b) As the ray emerges out of the prism perpendicular to the second face, the angle of emergence $\angle r_2=0$
So, using the relation,
$i+e=A+\delta$
where $\delta =$ angle of deviation,
$⟹{45}^{\circ }+0^{\circ }={30}^{\circ }+\delta$
$⟹\delta ={15}^{\circ }$
So, the angle of deviation for the given condition is ${15}^{\circ }$.