Animated Solution for Physics - Optics: A fish looking up through the water sees the outside world, contained in a circular horizon. If the refractive index of water is $4/3$ and the fish is $12\text{ cm}$ below the water surface, the radius of this circle (in cm) is

Imagine you are a fish swimming peacefully at a depth of $12\text{ cm}$ below the surface of a crystal-clear lake. When you look straight up, you don't see an endless expanse of the sky. Instead, the entire world above the water—the trees, the clouds, the sun—is compressed into a single, glowing circular window right above you.

This fascinating optical phenomenon is known as Snell's Window. But why does it happen, and what lies beyond this glowing circle? Let's dive into the physics of Total Internal Reflection (TIR) to uncover the mystery.

The Critical Angle

To understand Snell's Window, we must trace the path of light rays traveling from the water (a denser medium) into the air (a rarer medium). As light rays hit the water-air interface, they bend away from the normal due to refraction.

As the angle of incidence increases, the refracted ray bends closer and closer to the water surface. Eventually, at a specific angle called the critical angle (${\theta }_c$), the refracted ray grazes the surface perfectly at $90^{\circ }$.

If a ray hits the surface at an angle even slightly greater than ${\theta }_c$, it cannot escape the water. Instead, it acts as if it hit a perfect mirror and bounces right back down. This is Total Internal Reflection.

Because of this, the fish can only receive light from the outside world if the rays enter the water within a cone defined by this critical angle. The base of this cone on the water surface forms the circular horizon the fish sees.

Let's translate this beautiful geometry into mathematics. By looking at the right-angled triangle formed by the fish, the center of the circular window, and the edge of the window, we can relate the radius $R$ to the depth $h$.

Using basic trigonometry, the radius is given by:

We know from Snell's Law that the critical angle is determined entirely by the refractive index $\mu$ of the water:

Given that the refractive index of water is $\mu =\frac{4}{3}$, we can easily find the sine of the critical angle:

Here is where many students make a silly mistake—they substitute the sine value directly! However, our geometric formula requires $\tan {\theta }_c$.

If $\sin {\theta }_c=\frac{3}{4}$, we can imagine a right-angled triangle where the perpendicular is $3$ and the hypotenuse is $4$. By the Pythagorean theorem, the base is:

Therefore, the tangent of the critical angle is:

Now, we simply substitute this back into our master equation along with the depth $h=12\text{ cm}$:

And there we have it! The radius of the circular horizon is exactly $\frac{36}{\sqrt{7}}\text{ cm}$.

Notice how elegant this result is. The radius of Snell's Window is directly proportional to the depth of the observer. If the fish decides to dive deeper into the lake, its window to the world will expand, giving it a wider, albeit still compressed, view of the sky above.