Animated Solution for Physics - Optics: A ray of light passing through a prism ( $\mu =\sqrt{3}$ ) suffers minimum deviation. It is found that the angle of incidence is double the angle of refraction within the prism. Then, the angle of prism is ……… (in degrees).

Imagine a ray of light gracefully entering a glass prism. As it crosses the boundary from air to glass, it bends, travels through the prism, and bends again as it exits. This beautiful phenomenon is governed by Snell's Law. However, this problem presents a very special scenario: the prism is in a state of minimum deviation.

What does minimum deviation physically mean? It implies a state of perfect symmetry. The light ray travels exactly parallel to the base of the prism. Because of this symmetry, the angle at which the light enters the prism (the angle of incidence, $i$) is exactly equal to the angle at which it leaves (the angle of emergence, $e$). Furthermore, the angle of refraction $r$ at the first surface is exactly half of the prism's apex angle $A$. Mathematically, we write this as:

The problem provides a fascinating clue that acts as the key to unlocking the solution. We are told that the angle of incidence $i$ is exactly double the angle of refraction $r$ inside the prism. Let's express this mathematically:

Now, let's merge this clue with our knowledge of minimum deviation. Since we already established that $r=\frac{A}{2}$, we can substitute this into our given condition:

This simplifies beautifully to:

This is a profound realization! The angle at which the light strikes the prism is exactly equal to the angle of the prism itself.

Now, let's bring in the heavy machinery: Snell's Law. For a prism, the refractive index $\mu$ relates the angle of incidence to the angle of refraction at the first interface:

We are given that the refractive index $\mu =\sqrt{3}$. We also know that $i=A$ and $r=\frac{A}{2}$. Let's substitute all these pieces into Snell's Law:

To solve this equation, we need to dig into our mathematical toolbox and pull out a trigonometric identity. Specifically, the double angle formula for sine. We know that $\sin (2\theta )=2\sin \theta \cos \theta$. By applying this logic, we can expand $\sin A$:

Substituting this expansion back into our equation yields:

Look at the elegance of that equation! The $\sin \left(\frac{A}{2}\right)$ terms in the numerator and denominator cancel out perfectly, leaving us with a much simpler expression:

Dividing both sides by 2, we isolate the cosine term:

Now, we ask ourselves: for what angle is the cosine equal to $\frac{\sqrt{3}}{2}$? From standard trigonometric values, we know this angle is $30^{\circ }$. Therefore:

Multiplying by 2 gives us our final answer:

The prism angle is exactly $60^{\circ }$, revealing that we are dealing with a perfect equilateral prism. The seamless blend of optical physics and trigonometric identities makes this a truly elegant problem.