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Q.
At which angle the intensity of transmitted light is maximum when a polaroid sheet is rotated between two crossed polaroids?
Wave Optics
Solution:
Suppose $I_{0}$ be the intensity of polarised light after passing through the first polariser $P_{1}$. Then, the intensity of light after passing through second polariser will be
$I=I_{0} \cos ^{2} \theta$
where, $\theta$ is the angle between pass axes of $P_{1}$ and $P_{2}$. Since, $P_{1}$ and $P_{3}$ are crossed the angle between the pass axes of $P_{2}$ and $P_{3}$ will be $\left(\frac{\pi}{2}-\theta\right)$.
Hence, the intensity of light emerging from $P_{3}$ will be
$I =I_{0} \cos ^{2} \theta \cos ^{2}\left(\frac{\pi}{2}-\theta\right)$
$=I_{0} \cos ^{2} \theta \cdot \sin ^{2} \theta=\left(\frac{I_{0}}{4}\right) \sin ^{2} 2 \theta$
Therefore, the transmitted intensity will be maximum when $\theta=\pi / 4$.