Example

Question:

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

Solution:

Let \( I_0 \) be the intensity of polarised light after passing through the first polariser \( P_1 \). After passing through the second polariser \( P_2 \), the intensity is: \[ I = I_0 \cos^2\theta \] where \( \theta \) is the angle between the pass axes of \( P_1 \) and \( P_2 \).
For a third polariser \( P_3 \) crossed with \( P_1 \), and pass axis angle between \( P_2 \) and \( P_3 \) as \( \left( \frac{\pi}{2} - \theta \right) \): \[ I = I_0 \cos^2\theta \cos^2\left(\frac{\pi}{2} - \theta\right) \] \[ = I_0 \cos^2\theta \sin^2\theta \] \[ = \frac{I_0}{4}\sin^2 2\theta \] Therefore, transmitted intensity is maximum when \( \theta = \frac{\pi}{4} \).