Question

What do you understand by polarized light ? When a third polaroid is rotated between two crossed polaroids, then discuss the change in the intensity of the transmitted light.

Hint:

Malus's Law, \(I = I_{initial} \cos^2\theta\), is the fundamental principle for problems involving a series of polarizers. Always apply it sequentially, where the output intensity and polarization of one polaroid become the input for the next.

Answer 1

Step 1: Understanding Polarized Light:
An ordinary light wave is an electromagnetic wave where the electric field vector vibrates in all possible directions perpendicular to the direction of propagation. This is called unpolarized light.
Polarized light is light in which the electric field vector is restricted to vibrate in only a single plane. This plane is called the plane of polarization. The process of restricting the vibrations of light to a single plane is called polarization. A device used to produce polarized light, such as a polaroid, has a specific pass-axis, and it only allows the component of the electric field vector parallel to this axis to pass through.
Step 2: Two Crossed Polaroids:
Let the first polaroid be the polarizer and the second be the analyzer. When they are crossed, their pass-axes are perpendicular to each other (the angle between them is 90\(^{\circ}\)). If unpolarized light of intensity \(I_0\) is incident on the polarizer, the intensity of light transmitted through it is \(I_1 = I_0/2\). This polarized light of intensity \(I_1\) then falls on the analyzer. According to Malus's Law, the intensity of light transmitted through the analyzer (\(I_2\)) is given by \(I_2 = I_1 \cos^2\theta\), where \(\theta\) is the angle between the pass-axes of the polarizer and analyzer. Since the polaroids are crossed, \(\theta = 90^\circ\). \[ I_2 = I_1 \cos^2(90^\circ) = I_1 \times 0 = 0 \] Thus, no light is transmitted through two crossed polaroids.
Step 3: A Third Polaroid Between Two Crossed Polaroids:
Now, let's introduce a third polaroid between the polarizer (P1) and the analyzer (P2). Let the pass-axis of this third polaroid (P3) make an angle \(\theta\) with the pass-axis of the first polaroid (P1).
The intensity of light after passing through the first polaroid P1 is \(I_1 = I_0/2\).
This polarized light now falls on the third polaroid P3. The angle between the pass-axes of P1 and P3 is \(\theta\). The intensity of light transmitted through P3 is \(I_3\), given by Malus's Law: \[ I_3 = I_1 \cos^2\theta = \frac{I_0}{2} \cos^2\theta \] The light emerging from P3 is now polarized at an angle \(\theta\) with respect to the original polarization axis. This light then falls on the analyzer P2. The pass-axis of P2 is at 90\(^{\circ}\) to P1. Therefore, the angle between the pass-axes of P3 and P2 is \( (90^\circ - \theta) \). The final intensity of light transmitted through the analyzer P2 is \(I_{final}\), given by Malus's Law again: \[ I_{final} = I_3 \cos^2(90^\circ - \theta) = \left( \frac{I_0}{2} \cos^2\theta \right) (\sin^2\theta) \]
Step 4: Discussion of the Change in Intensity:
The final transmitted intensity is \( I_{final} = \frac{I_0}{2} \cos^2\theta \sin^2\theta \). Using the trigonometric identity \( \sin(2\theta) = 2\sin\theta\cos\theta \), we can rewrite this as: \[ I_{final} = \frac{I_0}{2} \left( \frac{\sin(2\theta)}{2} \right)^2 = \frac{I_0}{8} \sin^2(2\theta) \] When the third polaroid is rotated, the angle \(\theta\) changes.
The transmitted intensity is zero when \( \sin(2\theta) = 0 \), which occurs at \( 2\theta = 0^\circ \) or \( 180^\circ \), so \( \theta = 0^\circ \) or \( 90^\circ \). This means if the middle polaroid is aligned with either the first or the second polaroid, no light gets through.
The transmitted intensity is maximum when \( \sin^2(2\theta) = 1 \), which occurs at \( 2\theta = 90^\circ \), so \( \theta = 45^\circ \).
The maximum intensity is \( I_{max} = I_0/8 \).
Conclusion: As the third polaroid is rotated between the crossed polaroids, light is now transmitted. The intensity of the transmitted light varies from zero, increases to a maximum value of \(I_0/8\) when the third polaroid is oriented at 45\(^{\circ}\) to the first, and then decreases back to zero.