Question

Two polarisers $ P_1 $ and $ P_2 $ are placed in such a way that the intensity of the transmitted light will be zero. A third polariser $ P_3 $ is inserted in between $ P_1 $ and $ P_2 $, at the particular angle between $ P_1 $ and $ P_2 $. The transmitted intensity of the light passing the through all three polarisers is maximum. The angle between the polarisers $ P_2 $ and $ P_3 $ is:

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{8} \)
• D. \( \frac{\pi}{3} \)

Hint:

To maximize the transmitted intensity through multiple polarizers, the angles between the polarizers should be chosen to align with the conditions of maximum intensity based on the formula for light transmission.

Answer 1

The Correct Option is A

This section details the analysis of polariser configurations and the condition for maximizing transmitted light intensity with a third polariser, \( P_3 \), placed between \( P_1 \) and \( P_2 \).

1. Initially, when \( P_1 \) and \( P_2 \) are oriented at \( 90^\circ \) (\( \frac{\pi}{2} \) radians) to each other, the transmitted light intensity is zero, as orthogonal polarisers block all light.
2. Upon inserting \( P_3 \) between \( P_1 \) and \( P_2 \), Malus's Law is applied to determine the angle for maximum transmitted intensity through all three polarisers. Malus's Law states that the intensity \( I \) of light transmitted through a polariser at an angle \( \theta \) to the incident polarized light is \(I = I_0 \cos^2 \theta\), where \( I_0 \) is the initial intensity.
3. The intensity of light after passing through the first polariser, \( P_1 \), is \( I_0/2 \).
4. If \( \theta_1 \) represents the angle between \( P_1 \) and \( P_3 \), and \( \theta_2 \) represents the angle between \( P_3 \) and \( P_2 \), the intensity \( I \) after transmission through \( P_3 \) is given by: \(I = \left( \frac{I_0}{2} \right) \cos^2 \theta_1 \cos^2 \theta_2\).
5. To maximize the final intensity, a function of \( \theta_1 \) and \( \theta_2 \), the optimal condition occurs when \( \theta_1 = \theta_2 \). Given the constraint \( \theta_1 + \theta_2 = \frac{\pi}{2} \), this yields \( \theta_1 = \theta_2 = \frac{\pi}{4} \).

This setup maximizes the light intensity passing through all three polarisers. Consequently, the angle between polarisers \( P_2 \) and \( P_3 \) is \( \frac{\pi}{4} \).