CBSE Class 12 · 2016 · All India · Wave OpticsLong Answer

(a) Unpolarised light contains field vibrations in all directions perpendicular to propagation, with no preferred direction. Whatever the orientation of the polaroid's pass axis, it always transmits the same average component, namely $\frac{I_0}{2}$. Hence rotating the polaroid produces no change in transmitted intensity.
When sunlight is scattered by air molecules, the light scattered at $90^{\circ }$ to the incident beam is observed to be plane polarised: the molecular dipoles oscillating along the line of sight do not radiate towards the observer, so only the perpendicular component reaches the eye.
(b) After $P_1$: $I_1=\frac{I_0}{2}$. $P_2$ is at $60^{\circ }$ to $P_1$:

Now $P_3$ is rotated to make angle $\theta$ with $P_2$.
For $\theta =30^{\circ }:I_3=\frac{I_0}{8}{\cos }^{2}30^{\circ }=\frac{I_0}{8}\cdot \frac{3}{4}=\frac{3I_0}{32}$.
For $\theta =60^{\circ }:$ this makes $P_3$ perpendicular to $P_2$ ($60^{\circ }+...$, net $90^{\circ }$ from $P_2$), so