$A$ polaroid $(I)$ is placed in front of a monochromatic source. Another polaroid $(II)$ is placed in front of this polaroid $(I)$ and rotated until no light passes. $A$ third polaroid $(III)$ is now placed in between $(I)$ and $(II)$. In this case,will light emerge from $(II)$? Explain.

(N/A) Yes,light will emerge from the second polaroid $(II)$.
Let the intensity of light emerging from the first polaroid $(I)$ be $I_0$. Since polaroids $(I)$ and $(II)$ are crossed,the angle between their pass axes is $90^{\circ}$.
When the third polaroid $(III)$ is placed between $(I)$ and $(II)$ at an angle $\theta$ with the pass axis of $(I)$,the intensity of light emerging from $(III)$ is given by Malus' Law: $I_1 = I_0 \cos^2 \theta$.
The angle between the pass axes of $(III)$ and $(II)$ will be $(90^{\circ} - \theta)$.
Therefore,the intensity of light emerging from $(II)$ is $I_2 = I_1 \cos^2(90^{\circ} - \theta) = I_0 \cos^2 \theta \sin^2 \theta = I_0 (\sin \theta \cos \theta)^2 = I_0 \left(\frac{\sin 2\theta}{2}\right)^2 = \frac{I_0}{4} \sin^2(2\theta)$.
Since $\sin^2(2\theta) \neq 0$ for $0 < \theta < 90^{\circ}$,light will emerge from the second polaroid $(II)$.