State and explain Brewster’s law of polarization.

"The tangent of a polarizing angle is equal to the (relative) refractive index of the transparent medium on which light is incident." If θ_p is the polarizing angle and μ is the refractive index of the medium, then \(\mu=\tan\left(\theta_p\right)\)

Proof:

Suppose a beam of unpolarized light AB incident with polarizing angle θ_p one the surface of a transparent surface XY of a transparent medium of refractive index μ. The beam  AB is reflected along with BC (plane-polarized) and refracted along BD (unpolarized or partially plane polarized).

When the reflected beam is completely plane-polarized, the corresponding incident angle θ_p is called the polarizing angle. Let r be the angle of reflection.

From Snell's law, 

\(\mu=\frac{\sin\left(i\right)}{\sin\left(r\right)}\)

But, for polariizing angle, i = θ_p , the reflected ray are perpendicular to each other.

∴ \(\mu=\frac{\sin\left(\theta_p\right)}{\sin\left(r\right)}\) ------- (i)

Since, rays BC and BD are perpendicular to each other,

∴\(\theta_p+r=90^\circ\)
⇒ \(r=90^\circ-\theta_p\) 

With this value, equation (i) becomes

\(\mu=\frac{\sin(\theta_p)}{\sin(90-\theta_p)}=\frac{\sin(\theta_p)}{\cos(\theta_p)}\)

∴ \(\mu=\tan(\theta_p)\)

i.e., refractive indxe of a transparent medium is equal to the tangent of angle of polarization for the medium.