Problem 19 - University Physics with Modern Physics

Question

A ray of light is traveling in a glass cube that is totally immersed in water. You find that if the ray is incident on the glass-water interface at an angle to the normal larger than 48.7\(^\circ\), no light is refracted into the water. What is the refractive index of the glass?

Step-by-Step Solution

Answer

The refractive index of the glass is approximately 1.765.

Step 1: Understand the Problem

The problem involves a ray of light traveling through a glass cube in water. We are asked to find the refractive index of glass when total internal reflection occurs at an angle of 48.7°.

Step 2: Identify Relevant Concept

The critical angle problem requires using Snell's Law for refraction and the concept of total internal reflection. Total internal reflection occurs when the angle of incidence exceeds the critical angle.

Step 3: Calculate the Critical Angle

The critical angle \( \theta_c \) is given as 48.7°. The relationship between the refractive indices of glass and water is defined by \( n_{glass} \sin(\theta_c) = n_{water} \sin(90°) \).

Step 4: Apply Snell's Law

Snell's Law is \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \). Since \( \theta_2 = 90° \) at the critical angle, we have \( n_{glass} \sin(48.7°) = n_{water} \).

Step 5: Rearrange and Solve for the Refractive Index of Glass

Using the known refractive index of water \( n_{water} = 1.33 \), solve for \( n_{glass} \): \[ n_{glass} = \frac{1.33}{\sin(48.7°)}\]Calculate the value using \( \sin(48.7°) \approx 0.7536 \).

Step 6: Perform Calculation

Substitute \( \sin(48.7°) \) into the equation to find:\[n_{glass} = \frac{1.33}{0.7536} \approx 1.765\]

Key Concepts

Snell's Law

Snell's Law is a fundamental principle in optics that describes how light rays change direction as they pass from one medium to another. It is expressed with the equation \( n_1 \sin(\theta_1) = n_2 \sin(\theta_2) \), where:

• \( n_1 \) and \( n_2 \) are the refractive indices of the two different media.
• \( \theta_1 \) is the angle of incidence, and \( \theta_2 \) is the angle of refraction.

The refractive index is a measure of how much a material can bend light. Snell's Law helps predict the path of light as it travels from one material into another. When light enters a more optically dense medium, like from air to water, it bends towards the normal line. Conversely, it bends away when moving to a less dense medium.

total internal reflection

Total internal reflection is a phenomenon that occurs when a light ray traveling through a medium hits the boundary with a less dense medium at an angle greater than the critical angle. Under these conditions, the ray doesn't exit the medium; instead, it reflects entirely back into the original medium. It only happens when:

• The light is traveling from a denser to a less dense medium.
• The angle of incidence is greater than the medium's critical angle.

This concept is crucial in optic technologies like fiber optics, where light signals are kept inside the cable, preventing data loss.

critical angle

The critical angle is the minimum angle of incidence at which total internal reflection occurs. It is specific to the pair of media through which the light is traveling and can be calculated using Snell's Law:\[\theta_c = \arcsin\left( \frac{n_2}{n_1} \right)\]Where:

• \(\theta_c\) is the critical angle.
• \(n_1\) is the refractive index of the denser medium.
• \(n_2\) is the refractive index of the less dense medium.

In cases like the one we examined, light exceeding the critical angle undergoes total internal reflection, helping applications like fiber optic communication by keeping the light within the medium.

refractive index

The refractive index is a dimensionless number that describes how light propagates through a medium. When light passes from air (refractive index ≈ 1) into another material, like water or glass, it slows down, and its direction changes. The amount of bending or refraction depends on the refractive indices of the involved materials.

• Refractive Index (n): \( n = \frac{c}{v} \)
• \(c\) is the speed of light in a vacuum.
• \(v\) is the speed of light in the material.

The refractive index directly influences phenomena like the critical angle. For example, the exercise highlights how knowing the refractive index of water helped deduce that of glass when total internal reflection was involved.