Question

In between two crossed polaroids A and B, a third polaroid C is placed in such a way that its polarising axis makes an angle \( \theta \) from the polarising axis of the polaroid A. If the intensity of the transmitted light from the polaroid A is \( I_0 \), then find out the intensity of the polarised light transmitted from the polaroid B. For which angle \( \theta \) the intensity of the transmitted light will be maximum?

Hint:

When polaroids are crossed, the intensity of transmitted light is maximized when the polarizing axis of the middle polaroid is at \( 45^\circ \) to the axes of the first and last polaroids.

Answer 1

Intensity of Polarized Light through Crossed Polaroids:
Let’s consider the configuration where polaroids A and B are crossed, meaning their polarizing axes are at \( 90^\circ \) to each other. A third polaroid C is inserted between them, with its polarizing axis making an angle \( \theta \) with the polarizing axis of polaroid A. The intensity of light after passing through polaroid A is \( I_0 \). 
1. Intensity of light after passing through polaroid A: The intensity of light after passing through polaroid A will be: \[ I_1 = I_0 \cos^2 \theta, \] where \( \theta \) is the angle between the polarizing axis of A and C. 
2. Intensity of light after passing through polaroid B: The intensity of light transmitted through polaroid B will depend on the angle between the polarizing axis of C and B. Let this angle be \( \theta' \). The intensity of the transmitted light from B is given by: \[ I_2 = I_1 \cos^2 \theta' = I_0 \cos^2 \theta \cos^2 \theta'. \] Since polaroids A and B are crossed, \( \theta' = 90^\circ - \theta \). Thus, we have: \[ I_2 = I_0 \cos^2 \theta \sin^2 \theta. \] 
3. Maximum Intensity: To find the angle for which the intensity of the transmitted light is maximum, we differentiate \( I_2 \) with respect to \( \theta \) and set it equal to zero: \[ \frac{dI_2}{d\theta} = I_0 \left( 2 \cos \theta \sin \theta . \sin \theta . \cos \theta \right) = 0. \] The solution to this is: \[ \theta = 45^\circ. \] Hence, the intensity of the transmitted light will be maximum when the angle \( \theta \) between the polarizing axes of A and C is \( 45^\circ \).