Discuss the intensity of transmitted light when a polaroid sheet is rotated between two crossed polaroids.

(N/A) Let $I_{0}$ be the intensity of polarized light after passing through the first polarizer $P_{1}$.
When this light passes through the second polarizer $P_{2}$ (the rotating polaroid),the intensity $I$ is given by Malus' Law:
$I = I_{0} \cos^{2} \theta$
where $\theta$ is the angle between the pass axes of $P_{1}$ and $P_{2}$.
Since the first polarizer $P_{1}$ and the third polarizer $P_{3}$ are crossed,the angle between their pass axes is $\pi / 2$. If the angle between $P_{1}$ and $P_{2}$ is $\theta$,then the angle between $P_{2}$ and $P_{3}$ is $(\pi / 2 - \theta)$.
The intensity of light emerging from the third polarizer $P_{3}$ is:
$I_{final} = I \cos^{2}(\pi / 2 - \theta) = I_{0} \cos^{2} \theta \sin^{2} \theta$
Using the identity $\sin(2\theta) = 2 \sin \theta \cos \theta$,we can rewrite this as:
$I_{final} = I_{0} (\sin(2\theta) / 2)^{2} = (I_{0} / 4) \sin^{2}(2\theta)$
Thus,the transmitted intensity is maximum when $\sin^{2}(2\theta) = 1$,which occurs at $\theta = \pi / 4$ or $45^{\circ}$.