Question

Unpolarized light of intensity $32 \, W \, m^{- 2}$ passes through three polarizers arranged such that the transmission axes of the first and the last polarizer are at right angles. If the intensity of emerging light is $3 \, W \, m^{- 2}$ , then what is the angle (in degree) between the transmission axes of the first two polarizers?

Answer 1

Since unpolarized light is passing through the first polarizer, hence the intensity of light after crossing the first polarizer will be
$I_{1}=\frac{1}{2}I_{0}=16Wm^{- 2}$
Let us assume that the angle between the transmission axis of the first and second polarizer is $\theta $ , then from Malus law we can find out the intensity of light after it crosses the second polarizer.
$I_{2}=I_{1}cos^{2}\theta =16cos^{2}\theta $
Similarly, the intensity of light after crossing the third polarizer is
$I_{3}=I_{2}\left(cos\right)^{2}\left(90 ^\circ - \theta \right)=16\left(cos\right)^{2}\theta \left(sin\right)^{2}\theta $
$\Rightarrow I_{3}=16cos^{2}\theta sin^{2}\theta =3$
$\Rightarrow 4cos^{2}\theta sin^{2}\theta =\frac{3}{4}$
$\Rightarrow \left(sin\right)^{2}\left(2 \theta \right)=\frac{3}{4}$
$\theta =30^\circ $