Brewster’s Law – Definition, Formula, Derivation, Example, Uses

Brewster’s Law states that when unpolarized light strikes a surface at a specific angle (Brewster angle), the reflected light becomes polarized perpendicular to the plane of incidence. According to Brewster Law of polarization, the refractive index of a medium is equal to the tangent of the polarizing angle. This law finds applications in making polarizing filters for reducing glare in sunglasses and improving image quality in cameras, LCD screens, and optical devices by selectively transmitting polarized light.

In this article we will learn in detail about Brewster Law of Polarization, its formula, derivation, relationship with snell’s law and its application. We will also learn about Brewster angle and its relationship with critical angle in this article.

What is Brewster’s Law

Brewster Law states that the tangent of the polarizing angle of incidence of a transparent medium is equal to its refractive index. It was proposed by Sir David Brewster in 1811. He stated that when light hits a transparent surface, it is most polarized when the angle of incidence equals the angle where the reflected and refracted rays are perpendicular. This occurs because at the Brewster angle, the reflected and refracted rays are perpendicular to each other, resulting in the suppression of light polarized perpendicular to the surface. The value of Brewster angle depends on the nature of the transparent refracting medium and the wavelength of light used.

Brewster’s Law of Polarization

Brewster’s Law of Polarization states that when unpolarized light is incident on a transparent dielectric surface at a certain angle (called the Brewster angle), the reflected light becomes completely polarized parallel to the surface.

Polarization of Light

Polarization of Light is a phenomena due to which the oscillation of vectors associated to the light i.e. electric field vector and magnetic field are restricted to a single plane.

Brewster’s Law Formula

The formula for Brewster’s Law is

μ = tan θ_B

• where μ is the refractive index of the medium
• θ_B is the polarizing angle.

Derivation of Brewster’s Law

Consider the normal to the medium. The sum of angle on one side of the normal will be 180°

Let the incident angle be θ_B

From law of reflection, angle of incidence = angle of reflection = θ_B

According to Brewster’s Law, the reflected and refracted rays are perpendicular to each other. This implies that the angle of incidence i_p and the angle of refraction r_p at Brewster’s angle are complementary.

Now, on one side of normal sum of θ_B, angle of refraction and angle between polarized and unpolarized would be 180°

⇒ θ_B + 90° + r =180°

r = 90° – θ_B

Since Snell’s Law states that μ = sin(i) / sin(r) where, μ is refractive index of medium

Thus,

μ = sin θ_B/ sin(90– θ_B)

μ = sin θ_B/cos θ_B

⇒ μ = tan θ_B

Brewster Law relationship with Snell’s Law

From Snell’s law, the refractive index of the medium is

μ = sin i_p / sin r_p

μ = sin i_p / sin (90-i_p) = sin i_p / cos i_p = tan i_p ∵ (sin (90-a) = cos a)

so, μ = tan i_p = tan θ_B

μ = tan θ_B

Also, according to Snell’s Law describes the relationship between the angles of incidence and refraction for a wavefront traversing across the boundary between two media with different refractive indices is given by:

n₁ sin (i_p) = n₂ sin(r_p)

n₂/n₁ = sin (i_p)/ sin(r_p) = μ

Hence, μ = tan θ_B = n₂/n₁

Brewster’s Angle in Physics

Brewster angle is defined as the angle of incidence at which light with parallel polarization state is perfectly transmitted through the boundary with no reflection and light with perpendicular polarization state is reflected. The reflected light is completely polarized and has electric field oscillation confined to single plane only.

Mathematically Brewster Angle is given as

θ_B = tan^-1(n₂/n₁)

where n₁ and n₂ are the refractive indices of the two media involved

Derivation of Brewster’s Angle

Brewster angle can be calculated as,

μ = n₂/n₁

tan θ_B = n₂/n₁

θ_B = tan^-1(n₂/n₁)

θ_B = tan^-1(μ)

Brewster’s Angle and Critical Angle

Critical angle is the angle of incidence at which light is refracted at an angle of 90 degrees with respect to the normal, when it is incident on the boundary between two different media.

Since, μ = 1/Sin θ_c

Brewster Angle is given as θ_B = tan^-1(n₂/n₁)

Hence, θ_B = tan^-1(n₂/n₁) = tan^-1(cosec θ_c )

Brewster’s Law Experiment

To conduct a Brewster’s Law experiment, you’ll need:

• Laser or light source
• Polarizing filter
• Transparent medium (e.g., glass or acrylic)
• Protractor or angle measurement tool
• Surface with adjustable angle

Procedure:

1. Set up your light source and place a polarizing filter in front of it.
2. Place the transparent medium (glass or acrylic) on a surface with an adjustable angle.
3. Adjust the angle of the surface until the light beam from the source strikes the surface of the medium at various angles of incidence.
4. Use a protractor or angle measurement tool to measure the angle of incidence θ_i
5. Rotate the polarizing filter until the intensity of the light transmitted through the medium is minimized.
6. Record the angle of the polarizing filter (θ_B), which corresponds to Brewster’s angle.
7. Calculate the refractive index of the medium using the formula: μ = tanθ_B

By changing the angle of incidence and measuring the resulting Brewster angle, you can confirm Brewster’s Law, which states that the tangent of the Brewster angle equals the ratio of the refractive indices of the two media.

Observation of Polarization

Confirm that at Brewster’s angle, only p-polarized light is transmitted through the medium, while s-polarized light is predominantly reflected. This observation validates the polarization phenomenon described by Brewster’s Law.

The result of a Brewster’s Law experiment includes the determination of Brewster’s angle, the calculation of the refractive index of the medium, verification of Brewster’s Law, and confirmation of the polarization phenomenon at the interface between two media.

Applications of Brewster’s Law

Brewster’s Law has several practical applications in optics and technology due to its ability to selectively polarize light. Some of these applications include:

Polarizing Filters

Brewster’s Law is utilized in the manufacturing of polarizing filters. These filters are commonly used in photography, sunglasses, LCD screens, and other optical devices to block or selectively transmit polarized light.

Glare Reduction

Polarizing filters based on Brewster’s Law are extensively used in sunglasses and camera lenses to reduce glare from reflective surfaces such as water, glass, or snow. When light reflects off these surfaces, it becomes partially polarized parallel to the surface. By using polarizing filters aligned with Brewster’s angle, glare caused by horizontally polarized light can be effectively minimized.

Anti-Reflection Coatings

Brewster’s law is also used in the design of anti-reflection coatings for optical components such as lenses and windows. By applying a thin film of a material with a refractive index that is between the refractive indices of the two media, Brewster’s angle can be optimized to minimize reflections at the surface.

Ellipsometry

Brewster’s angle is employed in ellipsometry techniques for measuring the properties of thin films and surfaces. Ellipsometry is a non-destructive optical technique used to determine the thickness, refractive index, and other optical properties of thin films by analyzing the changes in polarization state of light reflected or transmitted from the sample surface.

Solved Examples on Brewster Law

Example: Determine the angle of refraction and polarization angle of the polarizer if the refractive index of the polarizer is 1.73.

Solution:

Given, Refractive index of the polarizer = 1.73

According to Brewster’s law, μ = tan θ_B = n₂/n₁

θ_B = tan^-1μ

θ_B = tan^-1(1.73) = tan-1 (√3)

θ_B = 60⁰ (angle of polarisation)

Since, i_p + r_p = 90⁰

60⁰ + r_p = 90⁰ (As θ_B and i_p are same)

r_p = 90⁰– i_p

r_p = 90⁰-60⁰

r_p = 30⁰ ( angle of refraction)

FAQs on Brewster Law of Polarization

What is Brewster’s Law?

Brewster’s Law states that when light hits a surface at a specific angle, called the Brewster angle, the reflected light becomes polarized.

What is Polarization of light?

Polarization of light refers to the orientation of the electric field vector associated with a light wave as it propagates through space.

How is Brewster’s Law applied in real-life scenarios?

Brewster’s Law is applied in polarizing filters, like those found in sunglasses and camera lenses, to reduce glare from reflective surfaces.

What is the significance of Brewster’s angle?

Brewster’s angle is significant because at this specific angle, light reflecting off a surface becomes perfectly polarized. This means it vibrates in a single direction, which is useful for reducing glare in sunglasses and enhancing image quality in cameras by selectively blocking unwanted light reflections.

Can Brewster’s Law be demonstrated practically?

Yes, Brewster’s Law can be practically demonstrated. By adjusting the angle of incidence of polarized light on a transparent medium, observe changes in transmitted light intensity. Identify Brewster’s angle, measure it, and calculate the medium’s refractive index using the Brewster’s Law equation. Validation is done by comparing the calculated refractive index with known values for the material.

How does Brewster’s Law relate to polarized light?

Brewster’s Law explains how light becomes polarized when it reflects off a surface at a specific angle called the Brewster angle. This phenomenon is crucial for creating polarized light, which is used in sunglasses and other optical devices.