Problem 13 - Master Resource Book in JEE Main Physics

Question

Two Nicol prisms are first crossed and then one of them is rotated through \(60^{\circ}\). The percentage of incident light transmitted is (a) \(\underline{1.25}\) (b) \(25.0\) (c) \(37.5\) (d) 50

Step-by-Step Solution

Answer

The percentage of incident light transmitted is 25%.

Step 1: Understanding Nicol Prisms and Light Transmission

Nicol prisms are used to polarize light. When two Nicol prisms are crossed, no light gets through. But when the second prism is rotated, some light is transmitted based on the angle of rotation. The formula that gives us the intensity of the transmitted light when the second Nicol prism is rotated by angle θ is given by Malus's law: \[ I = I_0 imes ext{cos}^2(θ) \] where, - \( I \) is the transmitted light intensity,- \( I_0 \) is the initial light intensity, and - \( θ \) is the angle of rotation of the second prism from the crossed position.

Step 2: Calculating Transmitted Light Intensity

Given that the second prism is rotated by \(60^{\circ}\), we need to calculate \( \text{cos}^2(60^{\circ}) \). Since the prisms are initially crossed, any transmitted light will be a percentage of the incident light:\[I = I_0 \times \text{cos}^2(60^{\circ})\]We know \( \text{cos}(60^{\circ}) = 0.5 \). Therefore,\[I = I_0 \times (0.5)^2 = I_0 \times 0.25\]Thus, \( I = 0.25 \times I_0 \) which means 25% of the incident light is transmitted.

Step 3: Selecting the Correct Answer

From the calculation in the previous step, the percentage of incident light transmitted when the second prism is rotated by \(60^{\circ}\) is 25%. This corresponds to choice (b).

Key Concepts

Nicol prisms

Nicol prisms are special optical devices that play a crucial role in the study and application of polarized light. They are made by cutting a natural crystal of calcite into two specific parts, then cementing them together at an angle. This construction allows the Nicol prism to separate unpolarized light into two polarized beams.
Nicol prisms are used to produce linearly polarized light by eliminating one of the beams. When unpolarized light, which normally vibrates in all directions perpendicular to its travel path, passes through a Nicol prism, only light vibrating in one direction is allowed through. This is useful in various scientific and practical applications where controlling light's direction is critical.

• Used in instruments requiring polarized light.
• Helps in reducing glare in photography and vision correction.

The behavior of light transmission with Nicol prisms becomes particularly interesting when two prisms are used together, as seen in experiments and exercises related to polarization.

Malus's law

Malus's law is fundamental to understanding how light behaves as it passes through polarized filters or devices such as Nicol prisms. Named after Étienne-Louis Malus, this law provides a mathematical relationship that predicts the intensity of light transmitted through a series of polarizing filters.
According to Malus's law, the intensity of polarized light after passing through a second polarizer is given by the formula:
\[ I = I_0 \times \text{cos}^2(\theta) \]
Here, \( I_0 \) is the intensity of light before it strikes the second polarizer, \( \theta \) is the angle between the light's initial polarization direction and the axis of the polarizer, and \( I \) is the intensity of the light after passing through.

• \(\theta\) impacts how much light travels through.
• An angle of 90° means no light is transmitted (crossed prisms).
• Smaller angles allow more light to transmit.

In practice, if two Nicol prisms are initially crossed, the transmitted light is nearly zero. Rotating one prism changes \(\theta\), and some light becomes visible again, which can be quantitatively predicted using this law.

light transmission

Light transmission through polarizing devices like Nicol prisms depends on several factors, primarily the angle of rotation and the initial light intensity. When two Nicol prisms are aligned in their crossed position, no light passes through — a vital step in understanding polarization experiments.
However, as one prism rotates, light transmission changes, following the concepts laid out in Malus's law. Specifically, when the prisms are adjusted, the transmission varies predictably based on the cosine of the angle squared. For example, rotating one prism by 60° from its crossed position results in a specific proportion of the original light intensity passing through:

• Initial transmission is zero at the crossed position (90°).
• Using \( \theta = 60° \), calculate \( \text{cos}^2(60°) = 0.25 \).
• This result means 25% of the initial light intensity is transmitted.

This information is particularly useful for applications measuring light properties and demonstrates the practical application of theoretical concepts like Malus's law in real-world scenarios managing light polarization.