Question

State Brewster’s law and show that when light is incident at polarizing angle the reflected and refracted rays are mutually perpendicular to each other.

Answer 1

Brewster’s law states that when unpolarized light is incident at a specific angle (θ_B​) on a transparent medium, the reflected light becomes completely polarized perpendicular to the plane of incidence. This special angle is called Brewster's angle. It is given by:

`tan i_p = "_1n_2= n_2/n_1` ....................(1)

where n₁ is the absolute refractive index of the surrounding and n₂ is that of the refracting medium. Figure shows a ray AO of unpolarised light incident on the interface separating two mediums. The degree of polarisation of the reflected ray OC varies with the angle of incidence and is a maximum for the angle of incidence equal to the polarising angle i_p of the pair of mediums. For all angle of incidence, the refracted ray OB is only partially polarised.

In figure, the angle of incidence is i_p , the angle of reflection is r’ and the angle of refraction is r. By Snell’s law,

`n_2/n_1= sin i_p/sinr`       ......(2)

From, Eqs. (1) and (2)

`tan i_p = sini_p/cos i_p= sini_p/sinr`

∴ cosi_p = sin r

Now, r' = i_p

∴ cos r' = i_p

∴ cos r' = sir r

∴ sin (90°-r') = sin r

∴ r = 90° - r'  or,  r' + r = 90°

∴ ∠ COB = 90°

At Brewster’s angle, the reflected ray is completely polarized, and the reflected and refracted rays are perpendicular to each other.