Two polaroids $P_1$ and $P_2$ are placed with their transmission axes strictly perpendicular to each

Question

Two polaroids $P_1$ and $P_2$ are placed with their transmission axes strictly perpendicular to each other. Unpolarised light of intensity $I_0$ is incident on $P_1$. A third polaroid $P_3$ is then kept precisely between $P_1$ and $P_2$ such that its transmission axis makes an angle of $60^{\circ}$ with that of $P_1$. The intensity of the transmitted light emerging through $P_2$ is

Professor PC Thomas Classes · Answer

(a) : The intensity of the initially unpolarised light transmitted completely through the first polaroid $P_1$ is $I_1 = \frac{I_0}{2}$. The angle between the transmission axes of $P_1$ and $P_3$ is $60^{\circ}$. By Malus's Law, the intensity transmitted through $P_3$ becomes $I_2 = I_1 \cos^2(60^{\circ}) = \left(\frac{I_0}{2}\right) \left(\frac{1}{2}\right)^2 = \frac{I_0}{8}$. Since $P_1$ and $P_2$ are completely crossed ($90^{\circ}$ apart), the angle between $P_3$ and $P_2$ is strictly $90^{\circ} - 60^{\circ} = 30^{\circ}$. The final intensity transmitted through $P_2$ is $I_3 = I_2 \cos^2(30^{\circ}) = \left(\frac{I_0}{8}\right) \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3I_0}{32}$.