Question

What is Brewster’s law? Derive the formula for Brewster’s angle.

Derive the formula for Brewster’s angle.

Answer 1

The tangent of the polarising angle equals the refractive index of the reflecting medium in comparison to the surrounding medium (₁n₂).

If θ_B = ₁n₂ = `n_2/n_1`

Here n₁ is the absolute refractive index of the surrounding and n₂ is that of the reflecting medium.

The angle θ_B is called the Brewster angle.

Consider a ray of unpolarized monochromatic light incident at an angle θ_B on a border between two transparent media, as illustrated in the figure below. Medium 1 has a lower refractive index (n₁) than Medium 2, which has a higher refractive index (n₂). Part of the incident light is refracted, and the rest.

The incident wave's electric field is perpendicular to the direction in which the incident light propagates. This electric field can be separated into two components: one parallel to the plane of the paper, represented by double arrows, and one perpendicular to the plane of the paper, represented by dots, both of equal magnitude. In general, reflected and refracted rays are partially polarised, which means that their magnitudes are not identical.

Sir David Brewster observed in 1812 that for a specific angle of incidence, θ_B, the reflected wave is totally plane-polarized, with its electric field perpendicular to the plane of the paper, but the refracted wave is partially polarised. This angle of incidence (θ_B) is known as the Brewster angle.

θ_B + θ_r = 90°  ....(1)

From snell’s law of refraction,

∴ n₁ sinθ_B = n₂ sin θ_r   ....(2)

From Eqs. (1) and (2), we have,

n₁ sin θ_B = n₂ sin (90° − θ_B)

= n₂ cos θ_B

∴ `n_2/n_1 = sintheta_B/costheta_B  = tan theta_B`

∴ `theta_B = tan^-1 (n_2/n_1)`  ....(3)

This is called Brewster’s law.