Question

a clear sheet of polarizing material is placed on top of a second, similar sheet so that their polarizing axes make an angle of 30° with each other. the ratio of the intensity of emerging light to incident unpolarized light is
1:3
1:4
3:4
3:8
1:2

Explanation:

Step1: Intensity after first polarizer

When unpolarized light of intensity \(I_0\) passes through a polarizer, the intensity of the light emerging from the first polarizer \(I_1=\frac{I_0}{2}\) (Malus's law for unpolarized light incident on a polarizer: the average of \(\cos^{2}\theta\) over \(0 - 2\pi\) for unpolarized light gives \(I = \frac{I_0}{2}\) as the light becomes linearly - polarized).

Step2: Intensity after second polarizer

Let the angle between the axes of the two polarizers be \(\theta = 30^{\circ}\). According to Malus's law \(I = I_{1}\cos^{2}\theta\). Substituting \(I_1=\frac{I_0}{2}\) and \(\theta = 30^{\circ}\) (so \(\cos\theta=\frac{\sqrt{3}}{2}\)), we get \(I_2=\frac{I_0}{2}\cos^{2}(30^{\circ})\).
Since \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\), then \(\cos^{2}(30^{\circ})=\frac{3}{4}\). So \(I_2=\frac{I_0}{2}\times\frac{3}{4}=\frac{3I_0}{8}\).

Answer:

\(3:8\)