Question

A plane polarized light is incident on a polariser with its pass axis making angle $?$ with $x$-axis, as shown in the figure. At four different values of $\theta, \theta = 8^{\circ}, 38^{\circ} , 188^{\circ} $ and $218^{\circ}$, the observed intensities are same. What is the angle between the direction of polarization and x-axis ?

• A. $98^{\circ}$
• B. $128^{\circ}$
• C. $203^{\circ}$
• D. $45^{\circ}$

Answer 1

The Correct Option is C

To solve this problem, we need to understand how the intensity of light behaves when a plane polarized light passes through a polarizer. The intensity of light passing through a polarizer is given by Malus's Law, which states that:

\(I = I_0 \cos^2(\theta - \phi)\)

where:

• \(I\): Intensity of the polarized light after passing through the polarizer.
• \(I_0\): Initial intensity of the polarized light.
• \(\theta\): Angle of the polarizer's axis with a reference direction (here, the x-axis).
• \(\phi\): Angle between the direction of polarization and the reference direction (x-axis).

We are given four different angles of the polarizer \((\theta = 8^{\circ}, 38^{\circ}, 188^{\circ}, 218^{\circ})\) where the intensity \(I\) is the same. According to Malus's Law, the calculated intensities being the same implies:

\(\cos^2(8^{\circ} - \phi) = \cos^2(38^{\circ} - \phi) = \cos^2(188^{\circ} - \phi) = \cos^2(218^{\circ} - \phi)\)

This means that the values inside the cosine function are equal modulo \(180^{\circ}\). Therefore, we have:

• \(8^{\circ} - \phi \equiv 38^{\circ} - \phi \pmod{180^{\circ}}\)
• \(38^{\circ} - \phi \equiv 188^{\circ} - \phi \pmod{180^{\circ}}\)
• \(188^{\circ} - \phi \equiv 218^{\circ} - \phi \pmod{180^{\circ}}\)

Evaluating these congruences for consistent intensity, we check each possible solution:

1. The difference between \(8^\circ\) and \(38^\circ\) is \(30^\circ\).
2. The difference between \(188^\circ\) and \(218^\circ\) is also \(30^\circ\).

This is possible if the angle \(\phi\) is such that it adjusts for a consistent \(30^\circ\) difference, suggesting symmetrical matching around:

1. \(\phi = 203^{\circ}\), placing it equally distributed along the axis from given angles of observation, confirming consistency.

Thus, the angle between the direction of polarization and the x-axis is \(203^\circ\).