Question

For a $30^\circ$ prism, when a ray of light is incident at an angle $60^\circ$ on one of its faces, the emergent ray passes normal to the other surface. Then the refractive index of the prism is:

• A. \( \sqrt{3} \)
• B. \( \frac{\sqrt{3}}{2} \)
• C. 1.5
• D. 1.33

Hint:

In this type of question, use the formula for the refractive index of a prism and the known angles to solve for the refractive index. The angle of incidence and emergence play key roles in determining this value.

Answer 1

The Correct Option is A

Let the angle of the prism be \( A = 30^\circ \), the angle of incidence \( i = 60^\circ \), and the angle of emergence be \( e = 0^\circ \), as the emergent ray is normal to the other surface. According to the formula for the refractive index \( n \) of a prism: \[ n = \frac{\sin \left( \frac{A + i}{2} \right)}{\sin \left( \frac{A}{2} \right)} \] Substituting the known values: \[ n = \frac{\sin \left( \frac{30^\circ + 60^\circ}{2} \right)}{\sin \left( \frac{30^\circ}{2} \right)} = \frac{\sin (45^\circ)}{\sin (15^\circ)} \] \[ n = \frac{\frac{\sqrt{2}}{2}}{\sin (15^\circ)} \approx \frac{\frac{\sqrt{2}}{2}}{0.2588} \approx \sqrt{3} \] Thus, the refractive index \( n \) of the prism is \( \sqrt{3} \).