Problem 89 - Master Resource Book in JEE Main Physics

Question

An unpolarised beam of intensity \(2 \alpha^{2}\) passes through a thin polaroid. Assuming zero absorption in the polaroid, the instensity of emergent plane polarised light is (a) \(2 a^{2}\) (b) \(a^{2}\) (c) \(\sqrt{2} a^{2}\) (d) \(\frac{a^{2}}{2}\)

Step-by-Step Solution

Answer

The correct answer is (b) \(\alpha^2\).

Step 1: Understanding the Problem

We have an unpolarized light beam with intensity \(2\alpha^2\) passing through a polaroid. We need to find the intensity of the light after it has been polarized. The polaroid does not absorb any light, it only polarizes it.

Step 2: Using the Malus's Law for Unpolarized Light

When unpolarized light passes through a polaroid, the intensity of the light is halved. The initial intensity is \(2\alpha^2\), so the intensity after passing through the polaroid is \(\frac{1}{2} \times 2\alpha^2\).

Step 3: Calculating the Intensity of Emergent Light

We substitute the initial intensity into the formula from Step 2: \[I = \frac{1}{2} \times 2\alpha^2 = \alpha^2.\]This gives the intensity of the emergent plane polarized light.

Key Concepts

Malus's Law

Malus's Law is a fundamental principle dealing with the intensity of light as it passes through a polarizer. When light interacts with a polarizer, its intensity changes based on the angle between the light's direction of polarization and the axis of the polarizer. However, keep in mind that this explanation assumes the light is initially polarized. Malus's Law is expressed with the formula:\[ I = I_0 \cos^2(\theta) \]Where:

• \(I\) is the intensity of light after passing through the polarizer.
• \(I_0\) is the initial light intensity before the polarizer.
• \(\theta\) is the angle between the light's initial polarization direction and the polarizer's axis.

In the case of unpolarized light, something interesting happens. Since unpolarized light has components vibrating in all directions perpendicular to its direction of travel, passing it through a polarizer results in half of its intensity being retained, irrespective of the angle \(\theta\). This is why unpolarized light of intensity \(2\alpha^2\) becomes \(\alpha^2\) after passing through a polaroid. It’s simplification of Malus's Law for unpolarized light.

Unpolarized Light

Unpolarized light is light that oscillates in multiple planes. Imagine the light wave as a snake slithering in different directions: upwards, sideways, and every way in between. This is what we call unpolarized, as there is no single defined direction of vibration.Common sources of unpolarized light include:

• Sunlight
• Light from a typical lamp
• Fire

When unpolarized light encounters a polarizer, like a polaroid or a pair of polarized sunglasses, the light emerging on the other side becomes polarized. Polarizers work by allowing only certain planes of light to pass through, usually reducing the intensity of the light by half. This is precisely why when unpolarized light with intensity \(2\alpha^2\) passes through a polaroid in our exercise, the resulting light intensity is halved, leading to an intensity of \(\alpha^2\). This splitting of light into components and allowing just one through is what transforms it into polarized light.

Intensity of Light

The intensity of light refers to the amount of light power per unit area, usually measured in watts per square meter (W/m²). It tells you how "strong" the light is and how much energy it conveys in a given space.Several factors can influence light intensity:

• The power of the light source
• The distance from the source (as light spreads, intensity decreases)
• Medium through which light passes (absorption can reduce intensity)

Understanding intensity is crucial when studying polarized and unpolarized light behaviors. For instance, when unpolarized light transitions through a polarizer, the intensity changes occur not because the light itself becomes weaker but because it’s being filtered to allow only one direction of light waves to pass. In the given exercise, the initial light has an intensity of \(2\alpha^2\), but after passing through a non-absorbing polaroid, its intensity decreases to \(\alpha^2\). Knowing how intensity behaves aids in understanding light’s interaction with materials and its propagation in different contexts.