Animated Solution for Physics - Optics: There is a small source of light at some depth below the surface of water (refractive index = $\frac{4}{3}$) in a tank of large cross-sectional surface area. Neglecting any reflection from the bottom and absorption by water, percentage of light that emerges out of surface is (nearly) [Use the fact that surface area of a spherical cap of height $h$ and radius of curvature $R$ is $2\pi Rh$]

Imagine you are a scuba diver resting at the bottom of a crystal-clear pool, looking straight up at the surface. You might expect to see the entire sky spread out above you, but physics has a different plan. Instead of a panoramic view, the world above is compressed into a bright, circular patch directly overhead. This fascinating phenomenon is known as Snell's Window.

Why does this happen? It all comes down to Total Internal Reflection (TIR). When light travels from a denser medium (like water) to a rarer medium (like air), it bends away from the normal. If the angle of incidence exceeds a specific threshold called the critical angle ($C$), the light cannot escape. Instead, it acts as if it hit a perfect mirror and bounces back into the water. Therefore, only the light rays that hit the surface within a cone of angle $2C$ can break free into the air.

To find the percentage of light that escapes, we need to think in three dimensions. A point source of light emits energy uniformly in all directions, creating an expanding spherical wavefront.

The light that successfully escapes the water is the portion that falls within our critical cone. Geometrically, the intersection of this cone with our imaginary spherical wavefront forms a spherical cap.

Therefore, the fraction of light that escapes is simply the ratio of the area of this spherical cap to the total surface area of the sphere.

Let's translate this geometry into mathematics. Suppose our imaginary sphere has a radius $R$. The total surface area of a complete sphere is well known:

The problem provides us with the formula for the area of a spherical cap:

Here, $h$ is the height of the cap. If we look at the cross-section of our cone and sphere, we can form a right-angled triangle. The hypotenuse is the radius $R$, the angle at the source is the critical angle $C$, and the adjacent side is the vertical distance from the source to the base of the cap, let's call it $h^{\prime }$.

Using basic trigonometry, we know that $h^{\prime }=R\cos C$. The height of the cap $h$ is simply the total radius $R$ minus this distance $h^{\prime }$.

Substituting this back into our cap area formula, we get:

Now, we can find the percentage of escaping light by taking the ratio:

Notice how beautifully the $2\pi R^2$ terms cancel out, leaving us with a remarkably simple expression:

To evaluate our final expression, we need the value of $\cos C$. The critical angle is defined by the refractive index $\mu$ of the medium:

We are given that the refractive index of water is $\mu =\frac{4}{3}$. Therefore:

Now, we use the fundamental trigonometric identity ${\sin }^{2}C+{\cos }^{2}C=1$ to find $\cos C$:

We are now ready to plug this value back into our percentage formula:

To find the numerical value, we approximate $\sqrt{7}\approx 2.645$:

Rounding to the nearest integer, we find that approximately $17\%$ of the light emerges from the surface.

This is a profound result! It means that a staggering $83\%$ of the light emitted by the underwater source is trapped beneath the surface, bouncing back down due to total internal reflection. This very principle of trapping light is what makes optical fibers possible, forming the backbone of our modern global telecommunications network.