Problem 34 - Arihant AIEEE Physics

Question

A glass prism of refractive index \(8 / 5\) is immersed in a liquid of refractive index \(4 / 3 .\) A ray of light incident at grazing angle on one face emerges at grazing angle on the other face of the prism. The angle of the prism is : (a) \(30^{\circ}\) (b) \(60^{\circ}\) (c) \(37^{\circ}\) (d) none of these

Step-by-Step Solution

Answer

Closest option is (b) \(60^{\circ}\).

Step 1: Understand the Problem

The problem is about a glass prism immersed in a liquid. We have the refractive indices of the prism \( \left(\frac{8}{5}\right) \) and the liquid \( \left(\frac{4}{3}\right) \). We need to find the angle of the prism given that light enters and exits at a grazing angle.

Step 2: Use the Snell’s Law

According to Snell’s law, \( n_1 \sin i = n_2 \sin r \), where \( n_1 \) and \( n_2 \) are the refractive indices, and \( i \) and \( r \) are the incidence and refraction angles. At grazing incidence or emergence, \( \sin i = 1 \) and \( \sin r = 1 \) since light travels along the boundary. Hence, \( n_{l} = n_{g} \sin A \), where \( n_{l} \) is the refractive index of the liquid, \( n_{g} \) is the refractive index of the glass, and \( A \) is the angle of the prism.

Step 3: Substitute the Known Values

Substitute the given refractive indices into the equation \( n_{l} = n_{g} \sin A \): \[ \frac{4}{3} = \frac{8}{5} \sin A \]

Step 4: Solve for the Angle A

Rearrange to find \( \sin A \): \[ \sin A = \frac{4}{3} \times \frac{5}{8} = \frac{5}{6} \]Now use the inverse sine function to find \( A \):\[ A = \arcsin \left( \frac{5}{6} \right) \]

Step 5: Calculate the Angle and Match Options

Using a calculator, find the value:\[ A \approx 56.44^{\circ} \]Since this is not exactly any of the provided options, choose the closest option.

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics used to describe the behavior of light as it passes from one medium into another. When light travels through different substances, like air to water, its speed changes, which causes the light to bend or refract. The law is mathematically expressed as:
\[ n_1 \sin i = n_2 \sin r \]
This equation tells us that the product of the refractive index (_1) of the first medium and the sine of the incident angle (i) equals the product of the refractive index (_2) of the second medium and the sine of the refraction angle (r).

• The refractive index () is a measure of how much a material can bend light.
• The incident angle (i) is the angle between the incoming light ray and the normal (an imaginary line perpendicular to the surface).
• The refraction angle (r) is the angle between the refracted light ray and the normal.

By ensuring the equation holds true, we can precisely predict how light will behave at the interface of two materials.

Refractive Index

The refractive index is a crucial concept that quantifies how much light bends when entering a medium. It's defined by the ratio of the speed of light in a vacuum to the speed of light in the material. The refractive index is denoted as \(n\). A higher refractive index implies that light travels slower in that medium.
In the exercise given:

• The refractive index of the glass prism is \(\frac{8}{5}\).
• The refractive index of the surrounding liquid is \(\frac{4}{3}\).

These values were used to calculate the angle of the prism by applying Snell's Law, taking into account the condition of light entering and exiting the prism at the grazing incidence, where light travels along the boundary, allowing for the simplification of the equation for finding the prism's angle.

Grazing Incidence

Grazing incidence is a term used when a ray of light travels almost parallel to the surface it intersects. This occurs when the incidence angle (i) is close to 90°, which means the ray barely grazes the surface.
In our exercise, the ray of light enters the prism at a grazing angle, and according to the situation described in the problem, it exits the prism at a grazing angle as well.

• When light is at a grazing incidence, \(\sin i = 1\) because the sine of 90° is 1. This simplifies calculations using Snell's Law.

Such a scenario is interesting because it allows for unique interactions with the prism's geometry, making grazing incidence crucial for calculations related to surfaces and angles, like finding the prism's angle here.

Angle of a Prism

The angle of a prism is the angular measure between its two refracting surfaces. In this problem, it is this angle that needs to be calculated. The challenge lies in determining this angle when light is entering and exiting the prism at grazing incidences.
The procedure involves:

• Utilizing the refractive indexes of the prism and surrounding medium.
• Applying Snell’s Law, considering the special condition that light is incident and emerges at grazing angles.

The equation \( n_{l} = n_{g} \sin A \) describes how these factors relate to the angle of the prism (A). Solving for \(A\) gives:
\[ \sin A = \frac{4}{3} \times \frac{5}{8} = \frac{5}{6} \]By calculating \(A = \arcsin\left(\frac{5}{6}\right)\), we find that the angle of the prism is approximately 56.44°, which is close to the options provided, indicating the angle's significance in the prism's design and its refractive properties.