Question

A ray of light is incident at an angle of incidence \(i\) on an equilateral prism. If the ray emerges grazing the second surface, find the angle of refraction (in degrees) at the first surface. Refractive index of the prism is \(\sqrt{2}\).

Hint:

In prism problems:
Grazing emergence always implies the internal angle equals the critical angle
For equilateral prisms, remember \(A = 60^\circ\)
Use geometry of the prism before applying Snell’s law

Answer 1

Correct Answer: 15

Concept:
For a prism:
The angle between the normals at the two refracting faces equals the prism angle \(A\).
For grazing emergence at the second surface, the angle of refraction inside the prism at the second surface equals the critical angle \(C\).
Snell’s law applies at each refracting surface. For an equilateral prism: \[ A = 60^\circ \]

Step 1: Determine the critical angle. Given refractive index: \[ \mu = \sqrt{2} \] Critical angle \(C\) is defined by: \[ \sin C = \frac{1}{\mu} \] \[ \sin C = \frac{1}{\sqrt{2}} \Rightarrow C = 45^\circ \]
Step 2: Use prism geometry. Let: \[ r_1 = \text{angle of refraction at first surface} \] \[ r_2 = \text{angle of incidence at second surface (inside prism)} \] For a prism: \[ r_1 + r_2 = A \] Since the ray emerges grazing the second surface: \[ r_2 = C = 45^\circ \] Hence: \[ r_1 = 60^\circ - 45^\circ = 15^\circ \]
Step 3: Final answer. The angle of refraction at the first surface is: \[ \boxed{15^\circ} \]