Two polarisers $P_1$ and $P_2$ are placed in such a way that the intensity of the transmitted light will be zero. A third polariser $P_3$ is inserted in between $P_1$ and $P_2$, at a particular angle between $P_2$ and $P_3$. The transmitted intensity of the light passing through all three polarisers is maximum. The angle between the polarisers $P_2$ and $P_3$ is:
Answer: (A) $\dfrac\pi4$
Zero transmission means $P_1$ and $P_2$ are crossed. If $P_3$ makes angle $\theta$ with $P_1$, it makes $90^\circ - \theta$ with $P_2$. By Malus's law, with $I_1$ the intensity after $P_1$:
$$I = I_1\cos^2\theta\cos^2(90^\circ - \theta) = I_1\cos^2\theta\sin^2\theta = \frac{I_1}{4}\sin^22\theta$$
This is greatest when $2\theta = 90^\circ$, so $\theta = 45^\circ$ and the angle between $P_2$ and $P_3$ is also $\dfrac\pi4$.
Solution by Sreeraj P, M.Sc Physics