Animated Solution for Physics - Optics: The angle of incidence at which reflected light is totally polarised for reflection from air to glass (refractive index $n$), is

When Light Becomes Perfectly Polarised

Have you ever wondered how polarised sunglasses work? They magically cut out the glare from the road or a lake. This isn't just a trick; it's a fundamental property of light interacting with surfaces, governed by a beautiful concept called Brewster's Law. Let's dive into the physics of how unpolarised light transforms into perfectly plane-polarised light.

When normal light, like sunlight, travels through space, its electric field vibrates in all possible directions perpendicular to its path. This is unpolarised light.

But when this light hits a boundary between two mediums—say, air and glass—something interesting happens. Part of the light is reflected, and part is refracted (bent) into the glass.

In 1815, a Scottish physicist named David Brewster made a fascinating observation. He noticed that at one specific angle of incidence, the reflected light was no longer vibrating in all directions.

Instead, it was completely plane-polarised, vibrating parallel to the surface! This special angle is now known as Brewster's angle or the polarising angle, denoted as $i_p$.

Brewster didn't just find the angle; he discovered the geometric secret behind it. He found that when light is incident at this exact polarising angle, the reflected ray and the refracted ray are perfectly perpendicular to each other. The angle between them is exactly $90^{\circ }$.

Mathematically, if the angle of reflection is $i_p$ (by the law of reflection) and the angle of refraction is $r$, then:

This simple geometric relationship is the key to unlocking the formula for Brewster's angle.

To connect this geometric secret to the properties of the glass, we use Snell's Law. Snell's Law relates the angle of incidence, the angle of refraction, and the refractive index $n$ of the medium:

Now, we substitute our geometric discovery into Snell's Law. Since $i_p+r=90^{\circ }$, we can write $r=90^{\circ }-i_p$.

Using basic trigonometry, we know that $\sin (90^{\circ }-\theta )=\cos \theta$. Applying this to our equation:

And since sine divided by cosine is tangent, the equation simplifies beautifully to:

This is the mathematical statement of Brewster's Law! To find the angle itself, we simply take the inverse tangent:

This elegant formula tells us that the polarising angle depends solely on the refractive index of the material. For glass with a refractive index of 1.5, Brewster's angle is about $56.3^{\circ }$.

Conclusion: The next time you put on a pair of polarised sunglasses and see the glare vanish, you'll know exactly why. The light bouncing off the road at Brewster's angle is horizontally polarised, and your vertically polarised lenses block it completely. It's a perfect harmony of geometry, trigonometry, and the wave nature of light!