# Brewster Law

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## What is Brewster Law?

We all know that all reflected lights don’t get polarized. In the plane of the medium, light is polarized at 90° to a plane that supports more reflection. When the light shines at a particular angle, it creates a huge impact on how polarized reflection is going to be. Therefore, Brewster’s law is here to guide us as to how the polarization varies with angle or simply the Brewster angle.

Now, we will understand Brewster law and Brewster angle in detail.

### State Brewster Law

So, we understood the brief of what is brewster law, now understand it in detail:

The maximum polarization occurs at an angle of 90° according to the Brewster law occurs between reflected and refracted rays. In 1811, a famous Scottish Physicist, Sir David Brewster, discovered this, and the law was named after him. Besides, the polarizing angle is known as Brewster's angle.

According to the Brewster law, we obtain the highest order of polarization of light by letting the rays come in contact with the surface of a transparent medium. Here, the refracted surface is perpendicular to the reflected ray, and a relationship develops between the polarizing angle ‘iP’ and the refractive index

### Point To Remember

The point to be noted is that the tangent of the polarizing angle is numerically equal to the medium's refractive index.

When we introduce a certain angle of incidence, a polarizing angle forms, the reflecting light gets entirely polarized. Here, a certain angle of incidence is known as a polarizing angle. On the surface of transparent material, the polarizing angle ‘iP’ attaches to the refractive index ‘μ’ of the material.

### Brewster Law Formula

We express the relation for polarising angle formula as μ = tan ip

Here,

μ refers to the refractive index of transparent medium

ip is the polarizing angle of incidence or a Brewster angle

When unpolarized light is incident on a transparent medium at any polarizing angle, then rays that transfer and reflect are vertical to each other.

Further, we see that as  = $\frac{(sinip)}{(Sin r)}$

Therefore, tan ip (polarizing angle) = $\frac{(sinip)}{(Sin r)}$

The above equation describes polarization by reflection.

### Brewster Angle Derivation

In his experiment, Brewster found that the reflected and refracted rays are orthogonal to each other when light is incident at a polarizing angle. Mathematically, the above statement can be written as;

ip+ 900 + r =1800

r = 900 - ip

From Snell's law, sini/sinr​ =μ

Sinip/​​sin(90−ip​) =μ

Or,

μ = tanip

We have proved the above statement that the tangent of the polarizing angle is numerically equal to the refractive index of the medium.

### Brewster's Angle

A Brewster angle is the angle at which an incident beam of unpolarized light gets reflected after the complete polarization is known as the Brewster angle or polarizing angle.

When an incident light with an electric field is parallel to the plane of incidence, you usually get zero reflection coefficients at an angle between 0 and 90°.

Because of this, the polarizing angle gets linearly polarized by having its electric field vectors parallel to the plane of the reflecting surface and perpendicular to the plane of incidence. At other angles, the reflected lights get partially polarized.

With the help of the Brewster law, the magnitude of Brewster's angle deviation depends upon the refractive indices of the involved optical channel and can be calculated.

So, what is the Brewster angle?

### What Is Brewster Angle?

B = arc tan(n2/n1)

Where n1 is the refractive index of the medium through which light propagates and n2 is the refractive index of the medium through which light reflects. It can also be shown that the sum of the angles in both media (relative to a direction for normal incidence) is 90°

By using the following equation, Brewster’s angle can be calculated as given below:

n = sin(qi)/sin(qr) = sin(qi)/sin(q90-i) = tan(qi)

Here,

n is the refractive index of the light reflecting medium

qr is the angle of refraction

qi is the angle of incidence

The above equation is useful for determining the refractive index of an unknown specimen such as opaque material with a high absorption coefficient for light transmission.

The critical Brewster angles for glass, water, and diamond are 57°, 53°, and 67.5°, respectively.

From the above arrangement, to obtain the disappearing reflection losses at the Brewster plate, the angle of incidence and Brewster's angle must be coincident, and light must be polarized, i.e., the direction of polarization should be in the plane of incidence (here, we have a drawing plane).

The light reflected from the surface at Brewster's angle produces shining effects. In modern lasers, Brewster's angle is an important concept to create linearly polarized light by reflections at the mirror surface of the laser cavity.

### Fun Fact

When you go hiking, you wear polarized sunglasses. When the weather is sunny, you might have noticed a beautiful scene of reflection on the lake.

The above two are polarized reflections. However, when you see through your sunglasses, you notice that the reflections fade. This happens because the sunglasses are polarized, which enables only polarized light to go through them. Also, you will find polarizers in monitors and TV screens to reduce the glare on sunny days.

Question 1: What are the Applications of Brewsters Law?

Answer: Polarized sunglasses are one of the general applications of Brewster's law. The concept of Brewster angle is applied to these glasses that can reduce glare emitted from the sun and the horizontal surface like water and road.

Even the photographers use this law to reduce the reflection from reflective surfaces by using polarized filters on their lenses.

In Brewster windows, both Brewster's law and Brewster's angles are applied. It is a glass that allows the windows to transmit 100% light.

Applications of Brewster’s law are found in gas lasers and solid-state lasers. In solid-state lasers, the end of the laser medium is cut at the brewster's angle to make a Brewster window.

Question 2: Find Brewster’s Angle of Light that Travels from Water (n = 1.33) into the Air?

Answer: We observed through experiments that refracted and reflected rays are at 90 degrees to each other when the light is incident at a polarizing angle. The tangent of a polarizing angle is numerically equal to the refractive index of the medium.

Now, let’s solve a question by applying this concept:

From the above question, the given refractive index is n₂ = 1.33

Now, applying Brewster’s law formula, we get:

Brewster’s angle = tan⁻¹(n₁/n₂​)

Brewster’s angle = tan⁻¹(1.5/1.33)

Thus, Brewster’s angle = 48.4°

So, Brewster’s angle is 48.4°.