Brewster’s law: Definition, Equation, Example, Brewster’s Angle

Brewster’s Law states the relationship of the waves of light at the maximum polarization angle of light. This relationship can be achieved by letting the ray fall on a surface of a transparent medium in such a way that the refracted ray makes an angle of 90° with the reflected ray. The reflected light is not all polarized. When it is in the reflection plane, the light is polarized at 90 degrees to the plane which makes it more likely to reflect. When one shines light at a particular angle, a huge impact is created on how the polarized reflection is going to turn out. Brewster’s law helps in understanding how the polarized reflection varies with the angle.

Key Terms: Brewster’s Law, unpolarized light, wavelength, reflection, refraction, polarized reflection, angle of light, Light, angle, polarization angle, Brewster’s angle

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Brewster’s Law

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According to Brewster’s Law,

This law also states that at a particular angle the polarized rays vanish completely on different glasses. The polarization angle is also called Brewster’s angle. It is an angle of incidence where the rays of light have p-polarization and is transmitted through a transparent dielectric surface without any reflection. The unpolarized light at this angle gets transmitted and the light gets reflected from the surface.

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Brewster’s Law Formula

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According to Brewster, the refractive index of the medium is equal to the tangent angle of polarization. 

So, as per the Brewster Law, the Brewster’s Law Formula can be derived as:

μ = tan i

Where, 

μ is the refractive index

i is the angle of polarization

According to Snell’s law,

μ = \(sin\ i \over sin\ r\) (1)

According to Brewster’s Law,

μ = tan i = \(sin\ i \over cos\ i\)(2)

Comparing (1) & (2)

cos i = sin r = cos (π/2 - r)

i = \(\pi \over 2\) - r

Or i + r = \(\pi \over 2\)

Therefore, the refracted and the reflected rays are at the right angle with each other as

i +r = \(\pi \over 2\) < ABC and is equal to \(\pi \over 2\)

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What is Brewster’s Angle?

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An angle where an incident beam of the unpolarized light is reflected after the complete polarization is called a polarizing angle or Brewster’s Angle. One usually gets zero reflection coefficients at an angle between 0°- 90°, if the incident light is parallel to the plane of the incidence in an electric field

\(\theta b\) = arc tan (n₂ /n₁)

Where,

n₁ is the refractive index of the medium from which light is propagated

n₂ represents the refractive index of the medium from which light gets reflected

Brewster’s angle can be calculated by:

n = \({sin(q_i) \over sin(q_r)} = {sin(q_i) \over sin(q_{90-i})} = tan (q_i)\)

Here, 

n = refractive index

q = angle of refraction

qi = angle of incidence

The above equation is helpful in determining the refractive index of an unknown compound such as opaque material with an absorption coefficient. 

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

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Relation between Brewster’s angle and Critical Angle

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Brewster’s angle can be given as:

θ_b = arc tan (n₂ /n₁)

Tan θ_b = (n₂ /n₁)

Critical angle can be given as:

θc = arc sin (n₁/n₂)

Sin θc = n₁/n₂

Sin θ_c=1/ (n₂ /n₁)

But, tan θ_p = n₂ /n₁

sin θc = \(1 \over tan\theta\ b\)

Hence, 

θc = arc sin \(1 \over tan\theta\ b\)

This is the relation between the Brewster’s Angle and Critical Angle. 

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Derivation of Brewster’s Angle 

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Brewster discovered that the refracted & reflected rays are both orthogonal to each other when the light is incident at an angle of polarization. If stated mathematically, it can be written as,

i_p + 90° + r = 180°

So, r = 90° - ip

Deriving it from Snell’s law,

\({Sin\ {ip} \over sin(90°- \ ip)} = \mu\)

Or, μ= tan ip

So the above statement along with Brewster’s angle derivation clearly shows that the angle of polarization is numerically equal to the refractive index of the medium.

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Applications of Brewster’s Law

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The applications of Brewster’s Law are as follows:

• Brewster Window is a 100% light-transmitting glass window that is used in gas lasers and also in solid-state lasers. In the case of the solid-state lasers, the ends are cut at Brewster’s angle to make Brewster’s Window.
• The Polarized sunglasses are one fine example of Brewster’s law. Brewster’s angle is used in the polarized sunglasses. It reduces the glare from the sun and the horizontal surfaces such as roads or water.
• Photographers also use this law to reduce the reflection by using the polarizing filter for the lens. 

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Things to Remember 

• The angle of polarization is also called Brewster’s angle.
• Brewster’s angle ‘n’ = \({sin(qi) \over sin(qr)}\)
• The light that gets reflected from the surface at Brewster’s angle makes a shining effect.
• The concept of Brewster’s angle is required for creating polarized light by reflections at the surface of the mirror of the laser cavity.
• The equation of Brewster’s angle is used for calculating the refractive index of any unknown specimen like opaque material with a high absorption coefficient of light transmission.
• The relationship between the Brewster’s Angle and Critical Angle is given as θc = arc sin (1/tanθb)
• The tangent of the polarizing angle will be numerically equal to the refractive index of the medium
• This law is used by photographers to reduce the reflection from reflective surfaces with the help of a polarizing filter for the lens.

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