Problem 48 - College Physics

Question

\(\bullet$$\bullet\) 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 greater 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 1.78.

Step 1: Understand the Problem

The given problem involves a glass cube immersed in water. Light is incident on the interface, and if the angle of incidence is greater than \(48.7^{\circ}\), total internal reflection occurs. We need to find the refractive index of the glass.

Step 2: Use Critical Angle Concept

Since no light is refracted into the water at angle \(48.7^{\circ}\), this angle is the critical angle. The condition for total internal reflection is met, meaning light transitions from a medium of higher refractive index (glass) to a medium of lower refractive index (water).

Step 3: Apply Snell's Law

According to Snell's Law, for the critical angle \(\theta_c\), \(n_{glass} \sin(\theta_c) = n_{water} \sin(90^{\circ})\). Here, \(n_{glass}\) is the refractive index of glass, and \(n_{water} = 1.33\) is the refractive index of water. \(\sin(90^{\circ}) = 1\).

Step 4: Plug in the Critical Angle

Substitute the critical angle \(\theta_c = 48.7^{\circ}\) and \(n_{water} = 1.33\) into the equation to find \(n_{glass}\):\[ n_{glass} \sin(48.7^{\circ}) = 1.33 \]

Step 5: Solve for Refractive Index of Glass

Calculate \(\sin(48.7^{\circ}) \approx 0.747\). Substitute into the equation:\[ n_{glass} \cdot 0.747 = 1.33 \]Solving for \(n_{glass}\):\[ n_{glass} = \frac{1.33}{0.747} \approx 1.78 \]

Key Concepts

Critical Angle

The critical angle is a fundamental concept in optics, particularly when discussing light traveling through different media. When light moves from a medium with a higher refractive index to one with a lower refractive index, there is an angle at which the light no longer refracts out of the medium but, instead, reflects entirely back into it. This specific angle is known as the critical angle.

The critical angle ( \( \theta_c \) ) is determined by the materials involved. For instance, light going from glass to water will have a particular critical angle, different from other material combinations. At the critical angle, the light skims the boundary and travels along the interface without entering the other medium.

To calculate the critical angle, we rely on Snell's Law. If the refractive index of the medium the light is exiting (say glass, \( n_{glass} \) ) is greater than the medium it's entering (water, \( n_{water} \) ), the basic formula for finding the critical angle is:

• Set the angle of refraction to \( 90^{\circ} \)
• Use \( n_{glass} \sin(\theta_c) = n_{water} \sin(90^{\circ}) \)

This ensures all light reflects internally beyond this angle.

Total Internal Reflection

Total internal reflection is an intriguing optical phenomenon that occurs when a wave hits a medium boundary at an angle greater than the critical angle such that the wave cannot pass through and, instead, reflects entirely back into the original medium. This concept is essential in applications like fiber optics, where light needs to be kept within glass or plastic fibers to transmit over distances without loss.

For total internal reflection to occur, two conditions must be met:

• The light must be traveling in a medium with a higher refractive index toward a medium with a lower refractive index.
• The angle of incidence must exceed the critical angle.

When these conditions are satisfied, no light escapes the initial medium; instead, it is "trapped" and continually reflects inside. This results in zero refracted light and an efficiency that can be extremely useful in technology dealing with light guidance and containment.

Fiber optics leverage this principle by bending light with minimal loss through long distances, enabling efficient data transmission.

Refractive Index

The refractive index is a measure that describes how light propagates through a medium. It is a dimensionless number, typically greater than 1, indicating how much the light will bend, or refract, when entering a material.

Mathematically, the refractive index, \( n \) , of a medium is calculated by the ratio of the speed of light in a vacuum ( \( c \) ) to the speed of light in the medium ( \( v \) ):

• \( n = \frac{c}{v} \)

For example, the refractive index of water is 1.33, meaning light travels 1.33 times slower in water than in a vacuum.

In the exercise involving a glass cube submerged in water, finding the refractive index of glass involved using known refractive indices and critical angle principles. Using Snell's Law, which links refractive indices to angles of incidence and refraction, we can calculate the unknown refractive indices when given certain conditions.

A higher refractive index indicates a greater bending effect on light entering the material, which is crucial in lenses, optical instruments, and understanding various natural phenomena.