Problem 2895 - JEE Physics

Question

The minimum angle of deviation of a prism of refractive index \(1.732\) is equal to its refracting angle. What is the angle of prism ? (A) \(45^{\circ}\) (B) \(30^{\circ}\) (C) \(60^{\circ}\) (D) \(40^{\circ}\)

Step-by-Step Solution

Answer

The angle of the prism is \(A = 30^{\circ}\).

Step 1: Understand the Prism Formula

We will be using the Prism Formula to solve this problem, which relates the angle of the prism (A), the angle of minimum deviation (δ), and the refractive index (n) of the prism material. The Prism Formula is given as follows: \(n = \frac{\sin{(A + δ)/2}}{\sin{(A/2)}}\) Since the minimum angle of deviation (δ) is equal to the refracting angle (A), we have: \(n = \frac{\sin{(A + A)/2}}{\sin{(A/2)}}\)

Step 2: Plug in the given refractive index

Now, we can plug in the given refractive index (1.732) in the Prism Formula and solve for the angle A. \(1.732 = \frac{\sin{(2A)/2}}{\sin{(A/2)}}\)

Step 3: Simplify and solve for the angle

We now have an equation we can solve for the angle A: \(1.732 = \frac{\sin{A}}{\sin{(A/2)}}\) Let's multiply both sides by \(\sin{(A/2)}\) to isolate the sinus of A: \(1.732 \cdot \sin{(A/2)} = \sin{A}\) Now we can look for an angle A that satisfies this equation.

Step 4: Check the given options

We will check each option to see which one satisfies the equation. (A) \(A = 45^{\circ}\) \(1.732 \cdot \sin{(45/2)} \approx 1.22\) (B) \(A = 30^{\circ}\) \(1.732 \cdot \sin{(30/2)} = 1.732\) Since option B satisfies the equation, the correct answer is: \(A = 30^{\circ}\)

Key Concepts

Prism Formula

The prism formula is essential to understanding how light behaves when it passes through a prism. This formula is used to determine the relationship between the angle of the prism, the angle of minimum deviation, and the refractive index of the prism's material. The formula is expressed as: \[ n = \frac{\sin{((A + \delta)/2)}}{\sin{(A/2)}} \] where \( n \) is the refractive index, \( A \) is the angle of the prism, and \( \delta \) is the angle of minimum deviation.

• The sine function \( \sin \) is used due to the nature of light refraction, which causes changes in speed and direction.
• This formula helps to calculate complex relationships in optical experiments, demonstrating the interaction of light with prisms.

In many practical scenarios, especially when working with triangular prisms, this formula allows us to solve for one unknown if the other variables are known. It's important to understand the components of this formula to accurately apply it in optical problems.

Refractive Index

The refractive index of a material defines how much light is bent, or refracted, when it enters the material. It is a critical factor in optics, as it affects how efficiently light propagates through different mediums. The refractive index \( n \) is calculated as the ratio of the speed of light in a vacuum to the speed of light in the material: \[ n = \frac{c}{v} \] where \( c \) is the speed of light in a vacuum, and \( v \) is the speed of light in the material.

• A refractive index greater than 1 indicates that light travels slower in that medium than in a vacuum.
• Materials with a higher refractive index bend light more than those with a lower refractive index.

In the exercise example, the prism has a refractive index of 1.732. This is indicative of a medium that bends light significantly, leading to the observed minimum deviation angle being equal to the refracting angle, a unique scenario in optics.

Angle of Minimum Deviation

The angle of minimum deviation (\( \delta \)) is a key concept in optics. It refers to the smallest angle through which light is deviated as it passes through a prism. This angle occurs when the light travels symmetrically through the prism, meaning the light beam enters and exits the prism at equal angles relative to the prism's faces.

• At the angle of minimum deviation, the path of light inside the prism is parallel to the base.
• This condition often simplifies calculations in optical problems since it relates directly to the prism's refracting angle.

In the problem scenario, the angle of minimum deviation is equal to the refracting angle of the prism. This symmetry implies that specific mathematical conditions are met, making the refractive index calculable through the prism formula. In many optical study setups, determining this angle aids in finding unknown prism properties or verifying material refractive indices.