$P_1, P_2, P_3$ are three Polaroid sheets, $P_1$ and $P_3$ are two crossed polaroids. What should be the angle between Polaroids $P_1$ and $P_2$, so that the value of $I'= I_o/4$

$θ=π/4$

The correct answer is Option (2) → $θ=π/4$

Let the transmission axis of P1 be 0°, P2 at θ, P3 at 90° (P1 and P3 crossed).

After P1 intensity = $I_0$ (given).

After P2: $I_1 = I_0 \cos^2\theta$.

Angle between P2 and P3 = $90^\circ - \theta$, so after P3:

$I' = I_1 \cos^2(90^\circ - \theta) = I_0 \cos^2\theta \sin^2\theta = I_0\left(\frac{\sin 2\theta}{2}\right)^2 = \frac{I_0}{4}\sin^2 2\theta$.

Require $I' = \frac{I_0}{4} \Rightarrow \sin^2 2\theta = 1 \Rightarrow 2\theta = 90^\circ \Rightarrow \theta = 45^\circ = \frac{\pi}{4}$.

Answer: $\theta = \frac{\pi}{4}$