Two polaroid $P_1$ and $P_3$ are placed such that their pass-axis are mutually perpendicular. Another polaroid $P_2$ is rotated between $P_1$ and $P_3$. For what angle θ between $P_1$ and $P_2$ the intensity of light emerging from $P_3$ will be maximum?

$\frac{π}{4}$

The correct answer is Option (3) → $\frac{π}{4}$

Intensity of light passing through a particular polarizer is -

$I=I_0\cos^2θ$ (θ → Angle between the light's polarization direction and pass axis)

Intensity after pass ($P_1$),

$I_1=\frac{I_0}{2}$

Intensity after pass ($P_2$),

$I_2=I_1\cos^2θ=\frac{I_0}{2}\cos^2θ$

Intensity after pass ($P_3$),

$I_3=I_2\cos^2(90-θ)$

$=\frac{I_0}{2}\cos^2θ\sin^2θ$

$=\frac{I_0}{2}×\frac{1}{4}×\sin^2(2θ)$

$=\frac{I_0}{8}\sin^2(2θ)$

for, Maximum intensity,

$\sin^2(2θ)=1$

$2θ=90$

$⇒θ=45°=\frac{π}{4}$