Problem 9 - University Physics with Modern Physics

Question

Light traveling in air is incident on the surface of a block of plastic at an angle of \(62.7^{\circ}\) to the normal and is bent so that it makes a \(48.1^{\circ}\) angle with the normal in the plastic. Find the speed of light in the plastic.

Step-by-Step Solution

Answer

The speed of light in the plastic is approximately \(2.53 \times 10^8\) m/s.

Step 1: Understand the Problem

We are given two angles and need to find the speed of light in plastic. The angles given are the angle of incidence (in air) of \(62.7^{\circ}\) and the angle of refraction (in plastic) of \(48.1^{\circ}\).

Step 2: Use Snell's Law

Snell's Law relates the angles of incidence and refraction to the indices of refraction of the two media. It is given by \(n_1 \sin \theta_1 = n_2 \sin \theta_2\). Here, \(n_1\) is the index of refraction of air (approximately 1) and \(n_2\) is the index of refraction of plastic which we'll solve for. \(\theta_1 = 62.7^{\circ}\) and \(\theta_2 = 48.1^{\circ}\).

Step 3: Calculate the Index of Refraction of Plastic

Rearrange Snell's Law to solve for \(n_2\): \(n_2 = \frac{n_1 \sin \theta_1}{\sin \theta_2}\). Substitute the known values: \(n_2 = \frac{1 \times \sin 62.7^{\circ}}{\sin 48.1^{\circ}}\). Calculate \(n_2 \approx \frac{0.884}{0.745} \approx 1.186\).

Step 4: Relate Index of Refraction to Speed of Light

The index of refraction is also defined as \(n = \frac{c}{v}\), where \(c\) is the speed of light in vacuum, approximately \(3 \times 10^8\) m/s, and \(v\) is the speed of light in the plastic. Rearrange this formula to solve for \(v\): \(v = \frac{c}{n}\).

Step 5: Calculate the Speed of Light in Plastic

Using the value of \(n_2\) from Step 3, \(v = \frac{3 \times 10^8}{1.186}\). Calculate \(v \approx 2.53 \times 10^8\) m/s.

Key Concepts

Index of Refraction

The index of refraction is a fundamental concept in optics that describes how much light slows down in a given medium compared to its speed in a vacuum.

• Symbol: The index of refraction is often represented by the letter "n."
• Formula: It's calculated using Snell's Law, which is expressed as: \[ n = \frac{c}{v} \]

In this formula: - "c" is the speed of light in vacuum, approximately **\(3 \times 10^8\)** m/s. - "v" is the speed of light in the material.A material with a higher index of refraction implies that light travels slower in that medium. For instance, in our exercise, the index of refraction for plastic was calculated to be approximately **1.186**. This means that light travels more slowly in plastic than in air, where the index of refraction is approximately 1.

Angle of Incidence

The angle of incidence is a key player in understanding how light behaves when it encounters a new medium. It is defined as the angle between the incoming light ray and the normal (an imaginary line perpendicular) to the surface at the point of incidence.

• The normal is an essential reference line, making it easy to measure angles relative to it.
• In our case, the angle of incidence in air was given as **62.7°**.

When light shines on a surface, this angle helps determine how much the light will bend or refract. Using Snell's Law and knowing the angle of incidence can help to calculate the angle of refraction and consequently the index of refraction of the new medium. This is crucial for understanding light's behavior upon entering a different environment, such as from air into plastic.

Speed of Light in Media

Light doesn't travel at the same speed everywhere. Its speed changes depending on the medium through which it propagates.

• In a vacuum, light travels at its maximum speed, approximately **\(3 \times 10^8\)** m/s.
• When light enters a medium like plastic, its speed is reduced.

The index of refraction helps us understand this change in speed. From Snell's Law and the calculation of the plastic's index of refraction, we found that the speed of light in plastic was approximately **\(2.53 \times 10^8\)** m/s. This reduction in speed is due to the interactions of light with the molecules in the medium, which cause it to slow down. Understanding these principles is essential for applications ranging from designing lenses to analyzing optical fibers in telecommunications.