Animated Solution for Physics - Optics: A ray of light passes from a denser medium to a rarer medium at an angle of incidence $i$. The reflected and refracted rays make an angle of $90^{\circ }$ with each other. The angle of reflection and refraction are respectively $r$ and $r^{\prime }$. The critical angle is given by

Have you ever looked at a straw resting in a glass of water and noticed how it seems to break or bend at the surface? This simple, everyday illusion is a gateway to one of the most profound and beautiful phenomena in physics: refraction. Today, we are going to embark on a thrilling journey through a classic optics problem that elegantly weaves together the laws of reflection, refraction, and the fascinating concept of the critical angle.

Imagine you are a photon of light traveling through a dense medium, perhaps a block of solid glass or a deep pool of water. Suddenly, you hit a boundary—an interface where your dense world meets a rarer medium, like the open air. At this exact moment, a split decision occurs. A portion of your energy bounces back into the dense medium; this is the reflected ray. The rest of your energy pushes through the boundary, bending as it enters the new, less restrictive environment; this is the refracted ray.

Let's assign some mathematical identities to our setup. We will call the refractive index of the denser medium ${\mu }_d$ and the refractive index of the rarer medium ${\mu }_r$. The angle at which our incident ray strikes the boundary, measured from the perpendicular normal line, is the angle of incidence, $i$. The reflected ray bounces off at an angle $r$, and the refracted ray bends at an angle $r^{\prime }$.

The problem hands us a beautiful geometric puzzle. First, we invoke the fundamental law of reflection, which states that the angle of incidence is always perfectly mirrored by the angle of reflection. Therefore, we can confidently state that $i=r$.

But here is the real twist: the problem reveals that the reflected ray and the refracted ray are exactly perpendicular to each other. They form a perfect $90^{\circ }$ angle. If we look at the straight, vertical normal line, we know that the angles on one side of it must sum up to a flat $180^{\circ }$.

So, if we add the angle of reflection $r$, the $90^{\circ }$ gap between the rays, and the angle of refraction $r^{\prime }$, they must equal $180^{\circ }$:

With a quick subtraction, this simplifies beautifully to give us the angle of refraction purely in terms of the angle of reflection:

Now, we bring in the heavy artillery: Snell's Law. This elegant equation bridges the two media, relating the angles of the rays to the refractive indices of the materials they travel through. At our interface, Snell's Law dictates:

Let's substitute the geometric relationships we just uncovered. We replace $i$ with $r$, and we replace $r^{\prime }$ with $(90^{\circ }-r)$:

Here, trigonometry offers us a neat trick. The sine of $(90^{\circ }-r)$ is mathematically identical to the cosine of $r$. Making this substitution, our equation evolves into:

By rearranging the terms to group the trigonometric functions on one side and the refractive indices on the other, we uncover a powerful ratio:

We have successfully linked the physical angles to the properties of the media. But the question asks for the critical angle, denoted as $C$. What exactly is a critical angle?

When light travels from a denser medium to a rarer medium, it bends away from the normal. If you keep increasing the angle of incidence, eventually the refracted ray will bend so much that it grazes perfectly along the boundary interface. The angle of incidence that causes this $90^{\circ }$ refraction is the critical angle. Mathematically, it is defined by the ratio of the refractive indices:

The final mathematical crescendo is right in front of us. We have two distinct expressions for the ratio $\frac{{\mu }_r}{{\mu }_d}$. One comes from our specific geometric setup, and the other is the universal definition of the critical angle. By equating them, we get:

To isolate the critical angle $C$, we simply take the inverse sine of both sides:

And there we have it! By meticulously following the path of the light ray, applying the laws of reflection and refraction, and leveraging a bit of clever geometry, we have unlocked the exact expression for the critical angle. Physics is not just about memorizing formulas; it is about seeing the hidden connections in the universe.