Question

Two polaroids are crossed to each other. Now, one of them is rotated through ${60}^{\circ }.$ What percentage of incident unpolarized light will pass through the system?

Answer 1

Let $I_o$ be the intensity of the incident unpolarized light on the system of the two polaroids. As the incident light is unpolarized, the intensity of light on being transmitted from the first polaroid will become
$I=I_{o}co{s}^2\theta =\frac{I_o}{2}$
If ${\theta }_1$ is the angle between the transmission planes of the two polaroids, then the intensity of light on passing through the second polaroid is given by
$I^{\prime }=Ico{s}^2{\theta }_1=\frac{I_o}{2}co{s}^2{\theta }_1$

Since the two polaroids were initially crossed, the angle between the transmission planes of the two polaroids on rotating through ${60}^{\circ }$ will become ${\theta }_1={90}^{\circ }-{60}^{\circ }={30}^{\circ }$.
$\therefore I^{\prime }=\frac{I_o}{2}\times co{s}^2{30}^{\circ }=\frac{I_o}{2}\times {\left(\frac{\sqrt{3}}{2}\right)}^2=0.375I_o$
Hence, the percentage of incident light that passes throgh the system.
$\frac{I^{\prime }}{I_o}\times 100=0.375\times 100=37.5\%$

Why this question? Tip: Half of the total intensity of unpolarized light passes through any polariod.