Problem 14 - 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 transmitted light is 25%.

Step 1: Understanding the Problem

We are given two Nicol prisms that are initially crossed, meaning they block all light. When one prism is rotated by an angle \( \theta = 60^{\circ} \), we are required to find the percentage of the incident light that is transmitted.

Step 2: Using Malus's Law

According to Malus's Law, when light passes through two polarizers, the intensity \( I \) of the transmitted light is given by \( I = I_0 \cos^2\theta \), where \( I_0 \) is the initial intensity of light and \( \theta \) is the angle between the axes of the polarizers.

Step 3: Applying the Angle

Substitute \( \theta = 60^{\circ} \) into Malus's Law: \( I = I_0 \cos^2(60^{\circ}) \). We need to calculate \( \cos^2(60^{\circ}) \).

Step 4: Calculating Cosine

The value of \( \cos(60^{\circ}) \) is \( \frac{1}{2} \). Squaring this value gives \( \cos^2(60^{\circ}) = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \).

Step 5: Finding Percentage of Transmitted Light

Since \( I = I_0 \cos^2(60^{\circ}) \) and \( \cos^2(60^{\circ}) = \frac{1}{4} \), the transmitted light intensity is \( \frac{1}{4} I_0 \). Therefore, the percentage of incident light that is transmitted is \( 25\% \).

Step 6: Conclusion

The percentage of the incident light transmitted when one prism is rotated by \(60^{\circ}\) is \( 25\% \).

Key Concepts

Nicol prisms

Nicol prisms are optical devices that play a crucial role in manipulating light. They are made from a special type of calcite crystal that splits an incoming light beam into two separate polarized beams. By eliminating one of these beams through a process called total internal reflection, a Nicol prism effectively allows only one beam to pass through, creating polarized light.
This ability to convert ordinary light into polarized light makes Nicol prisms extremely useful in a variety of applications, including photography, scientific experiments, and optical instruments. These prisms were named after their inventor, William Nicol, and remain highly valued in the study of light behavior and polarization.

polarization

Polarization refers to the orientation of light waves as they travel. Normally, light waves vibrate in multiple planes; however, when light is polarized, its waves are aligned to vibrate in just one plane. This is crucial for various optical technologies and helps in reducing glare in sunglasses and camera lenses.
Polarization occurs when light passes through materials like Nicol prisms or special filters known as polarizers. When two polarizers, such as Nicol prisms, are oriented perpendicularly (crossed), they block the passage of light completely. It’s fascinating to see how rotating one of the polarizers alters the amount of light that can pass through, demonstrating the principles of polarization.

transmitted light

Transmitted light refers to the light that successfully passes through a medium, like a Nicol prism or any polarizing material. In the context of polarization, transmitted light intensity can be calculated using Malus's Law, which provides a mathematical way to determine how much light gets through when it interacts with polarizing components.
To understand transmitted light using Malus's Law, consider two crossed Nicol prisms: when one prism is rotated by an angle, the intensity of the transmitted light is given by the equation:

• \( I = I_0 \cos^2\theta \), where \( I_0 \) is the initial intensity of the light, and \( \theta \) is the angle of rotation.

This means that as the angle changes, so does the intensity of the light passing through, showcasing a simple yet profound example of light behavior.