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

#### Solution

Let us consider two crossed polarisers P_{1} and P_{2}, with a polaroid sheet P_{3} placed between them.

Let *I*_{0} be the intensity of polarised light after passing through the first polariser P_{1}. If *θ* is the angle between the axes of P_{1}_{ }and P_{3}, then the intensity of the polarised light after passing through P_{3} will be I=I_{0}cos^{2}θ.

As P_{1}_{ }and P_{2} are crossed, the angle between the axes of P_{1}_{ }and P_{2} = 90°.

∴ Angle between the axes of P_{2} and P_{3}_{ }= (90°−θ)

The intensity of light emerging from P_{2} will be given by

I=[I_{0}cos^{2}θ]cos^{2}(90°−θ)

⇒I=[I_{0}cos^{2}θ]sin^{2}θ

`=>I=I_0/4(4cos^2thetasin^2theta)`

`=>I=I_0/4(2sinthetacostheta^2)`

`=>I=I_0/4sin^2theta`

The intensity of polarised light transmitted from P_{2} will be maximum when

sin2θ=maximum=1

⇒sin2θ=sin90°

⇒2θ=90°

⇒θ=45°

Also, the maximum transmitted intensity will be given by

`I=I_0/4`