State Brewster's law. Prove that if light is incident at the angle of polarization the reflected and refracted rays are mutually perpendicular.

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Solution

Brewster's law: The refractive index of a medium is equal to the tangent of angle polarization i.e.,
$μ = t a n i_{p}$
Let a light ray is incident at the angle of polarization $i_{p}$ , to the medium of refractive index $μ$ .
$∴$ By Snell's law, $μ = \frac{s i n i_{p}}{s i n r}$ .........(i)
Where, r is the angle of refraction.
But, $μ = t a n i_{p}$ ........(ii)
From equations (i) and (ii), we get
$t a n i_{p} = \frac{s i n i_{p}}{s i n r}$
or $\frac{s i n i_{p}}{s i n r_{p}} = \frac{s i n i_{p}}{s i n r}$
$∴ s i n (90^{o} - i_{p}) = s i n r$
or $90^{o} - i_{p} = r$
or $i_{p} + r = 90^{o}$
But we have, $i_{p} + r + ∠ C B D = 180^{o}$
or $90^{o} + ∠ C B D = 180^{o}$
or $∠ C B D = 90^{o}$
Thus, reflected ray and refracted ray are mutually perpendicular.

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