Question

Find an expression for intensity of transmitted light when a polaroid sheet is rotated between two crossed polaroid. In which position of the polaroid sheet will the transmitted intensity be maximum?

Answer 1

when a light ray which is vibrating in every direction passed through a polaroid, the motion or vibration gets aligned in a single direction. We can use the law of Malus in this question to get the expression.

Complete step by step answer
Crossed polaroid is an arrangement where the polarizing direction of two polaroid are mutually perpendicular. First one is called ‘polarizer’ and second one is ‘analyzer’. This case happens when two polaroid are present and the second one is rotated through the right angle such that no light goes through.
According to Malus law, if $\theta$ is the angle between analyzer and polarizer and a plane polarized light of intensity ${I_0}$falls on an analyzer then ,the emerging light will have intensity given by
$I = {I_0}{\cos ^2}\theta$
Now, when a polarized sheet is rotated between crossed polaroid then,
Assume intensity of unpolarized wave be ${I_0}$ and intensity of wave from second polaroid be ${I_1}$ and from the final polaroid intensity be ${I_f}$ then,
${I_1} = {I_0}{\cos ^2}\theta$
${I_f} = {I_1}{\cos ^2}(90 - \theta )$ as the first and third polaroid are crossed
Substituting the value final intensity becomes,
${I_f} = {I_0}{\cos ^2}\theta {\sin ^2}\theta = \dfrac{{{I_0}(4{{\cos }^2}\theta {{\sin }^2}\theta )}}{4} = \dfrac{{{I_0}}}{4}{(\sin 2\theta )^2}$this is the required expression
To get maximum transmitted intensity $\sin 2\theta = 1 \Rightarrow 2\theta = \dfrac{\pi }{2} \Rightarrow \theta = \dfrac{\pi }{4}$
When all polaroid and polaroid sheets are kept at $\dfrac{\pi }{4}$ with each other they will have maximum transmitted intensity.

Additional information
1. Polaroids are used in several ways in daily life –
2. They are used to view three -dimensional pictures.
3. Polaroids are set on the cover of headlights and windscreen of cars. This helps to avoid bright light from cars coming from the opposite direction.
4. They are used to control light intensity entering trains or planes.

Note:
We can see that when the polarizer and analyzer are perpendicular the transmitted light intensity will be zero.This can be mathematically proven by putting the value of angle as right angle and the cosine of this angle will be zero.