Animated Solution for Physics - Optics: A light ray is incident perpendicular to one face of a $90^{\circ }$ prism and is totally internally reflected at the glass-air interface. If the angle of reflection is $45^{\circ }$, we conclude that for the refractive index $n$ as

Imagine a beam of light traveling through the air and striking the vertical face of a right-angled glass prism. Because the light ray hits this first face perfectly perpendicularly (at an angle of incidence of $0^{\circ }$), it doesn't bend. It marches straight through the glass, completely undeviated, until it encounters the slanted hypotenuse of the prism.

This is where the magic happens. Instead of passing through the hypotenuse and refracting back into the air, the light ray is perfectly reflected back inside the glass. This phenomenon is known as Total Internal Reflection (TIR).

TIR is one of the most fascinating and useful phenomena in optics, responsible for everything from the sparkle of diamonds to the high-speed data transmission in fiber optic cables. For TIR to occur, two strict conditions must be met:

1. The light must be traveling from a denser medium (like glass) towards a rarer medium (like air). 2. The angle of incidence at the boundary must be strictly greater than a specific threshold known as the critical angle ($C$).

Mathematically, this crucial condition is written as:

Let's look closely at the geometry of our prism. The problem states that the angle of reflection at the hypotenuse is $45^{\circ }$. According to the fundamental Law of Reflection, the angle of incidence must perfectly equal the angle of reflection. Therefore, our angle of incidence $i$ is exactly $45^{\circ }$.

Substituting this into our TIR condition, we get our master inequality:

To solve for the refractive index $n$, we need to relate it to the critical angle. We do this by taking the sine of both sides of our inequality. Since the sine function is strictly increasing between $0^{\circ }$ and $90^{\circ }$, the direction of the inequality remains unchanged:

From Snell's Law, we know that the sine of the critical angle for a glass-air interface is given by $\sin (C)=\frac{1}{n}$. We also know from standard trigonometry that $\sin (45^{\circ })=\frac{1}{\sqrt{2}}$. Let's carefully substitute these values into our inequality:

We are almost there! We have a simple algebraic inequality. To isolate the refractive index $n$, we take the reciprocal of both sides. Remember the golden rule of algebra: when you take the reciprocal of positive numbers in an inequality, you must flip the inequality sign.

Reversing the sign, we arrive at our final, elegant result:

This tells us that for the prism to trap the light ray and force it to reflect internally at a $45^{\circ }$ angle, the glass must have a refractive index strictly greater than $\sqrt{2}$ (approximately $1.414$). If the refractive index were any lower, the light would simply refract and escape into the air. This beautiful interplay of geometry and optical physics is a classic, high-yield concept for JEE!