Question

The refractive index of prism is µ = √3 and the ratio of the angle of minimum deviation to the angle of prism is one. The value of angle of prism is ______^o.

Answer 1

Correct Answer: 60

For $\delta_\text{min}$:

\[ i = e \quad \text{and} \quad r_1 = r_2 = \frac{A}{2} \]

\[ \delta_\text{min} = 2i - A \]

Given, $\frac{\delta_\text{min}}{A} = 1$:

\[ \frac{2i - A}{A} = 1 \]

Simplifying:

\[ 2i - A = A \implies 2i = 2A \implies i = A \]

Using Snell's law:

\[ 1 \cdot \sin i = \mu \cdot \sin r \implies \sin i = \mu \cdot \sin \left(\frac{A}{2}\right) \]

Substituting $i = A$:

\[ \sin A = \mu \cdot \sin \left(\frac{A}{2}\right) \]

Expanding $\sin A$:

\[ 2 \sin \frac{A}{2} \cos \frac{A}{2} = \sqrt{3} \cdot \sin \frac{A}{2} \]

Dividing by $\sin \frac{A}{2}$:

\[ 2 \cos \frac{A}{2} = \sqrt{3} \implies \cos \frac{A}{2} = \frac{\sqrt{3}}{2} \]

\[ \frac{A}{2} = 30^\circ \implies A = 60^\circ \]