Question

Two identical television signals are sent between two cities that are \(400.0 \mathrm{km}\) apart. One signal is sent through the air, and the other signal is sent through a fiber optic network. The signals are sent at the same time but the one traveling through air arrives \(7.7 \times 10^{-4}\) s before the one traveling through the glass fiber. What is the index of refraction of the glass fiber?

Short answer

The index of refraction of the glass fiber is approximately 1.575.

Step-by-step solution

Understand the Problem

We are given a scenario where two identical signals are sent, one through the air and the other through a glass fiber, over a distance of 400 km, with the one through air arriving 0.00077 s earlier. We need to find the index of refraction of the glass fiber.

Use the Speed of Light in Air

The speed of light in air is very close to the speed of light in vacuum, approximately \[c = 3.00 \times 10^8 \text{ m/s}.\]This speed will be used to calculate the time taken by the signal through the air.

Calculate the Time for Air Travel

Using the speed of light, the time taken for the signal to travel in air can be calculated using the formula:\[ t_{ ext{air}} = \frac{d}{c}, \]where \(d = 400.0 \times 10^3 \text{ m} \). So, \[ t_{ ext{air}} = \frac{400.0 \times 10^3}{3.00 \times 10^8} \approx 0.00133 \text{ s}.\]

Determine the Time for Fiber Optic Travel

Since the signal through air arrives earlier by 0.00077 s, the time for the fiber optic travel is:\[ t_{ ext{fiber}} = t_{ ext{air}} + 7.7 \times 10^{-4} \approx 0.00133 + 0.00077 = 0.0021 \text{ s}.\]

Use the Index of Refraction Formula

The speed of light in the fiber optic, \( v \), is related to the index of refraction, \( n \), by:\[ v = \frac{c}{n}.\]Using the relation \( t = \frac{d}{v} = \frac{d \cdot n}{c}\),we can find \( n \) from \( t_{\text{fiber}} \):\[ t_{\text{fiber}} = \frac{d \cdot n}{c}, \]\[ n = \frac{t_{\text{fiber}} \cdot c}{d}. \]

Calculate the Index of Refraction

Substitute the known values:\[ n = \frac{0.0021 \cdot 3.00 \times 10^8}{400.0 \times 10^3} \approx 1.575.\]

Key concepts

Speed of Light

The speed of light is a fundamental constant in physics. It is the fastest speed at which information and energy can travel. Light moves at a speed of approximately \(3.00 \times 10^8\) meters per second (m/s) in a vacuum. This almost unimaginable speed means light can circle the Earth more than seven times in just one second!

Due to its fundamental nature, the speed of light is not just relevant to optics or electromagnetism but plays a crucial role in the theory of relativity and various technological applications.

In the context of the exercise, understanding the speed of light helps us calculate how quickly a signal sent through air arrives at its destination. The light traveling through air moves almost as fast as it would in a vacuum, allowing us to approximate its speed as the speed of light itself for practical problems.

Optical Fiber

An optical fiber is a thin, flexible strand of glass or plastic that efficiently transmits light from one end to the other. This technology relies on the principle of total internal reflection, allowing light to bounce down the length of the fiber with minimal loss.

Optical fibers are integral to telecommunications and internet infrastructure, enabling high-speed data transmission across vast distances with little degradation of the signal. They offer significant advantages over traditional metal cables, such as copper, in terms of bandwidth and signal integrity.

Within the original problem, we dealt with a signal traveling through a glass fiber optic cable over a significant distance. The scenario demonstrated how the fiber, despite being slower than air in terms of light speed due to its physical properties, is a crucial tool in modern long-distance communication.

Signal Transmission

Signal transmission refers to the process of sending a signal from one location to another, which can be done using various media, including air, optical fibers, and cables. The speed at which the signal travels depends on the medium's properties.

In the case of the given exercise, the television signal is transmitted through two different media: through the air and through an optical fiber. Each of these has unique characteristics that affect the time it takes for the signal to travel 400 km. Light travels faster through the air compared to the fiber due to the lower index of refraction. This results in the signal arriving at the destination sooner when sent through the air.

Optical fibers, while slower than the speed of light in air, are indispensable in signal transmission because they support long distances and high data bandwidth.

Physics Problem Solving

Physics problem solving involves breaking down complex scenarios into manageable calculations like what we've done with the light travelling through air and fiber optic cable.

By understanding the basic principles, such as speed, distance, and time, along with the properties of different media, we can systematically approach and resolve problems like the one given.

• First, identify and understand the problem.
• Use known physics equations and constants, such as the speed of light.
• Perform necessary calculations to find unknown variables, like the time difference or index of refraction.
• Verify each step to ensure accuracy.

In the problem provided, by understanding how light speed and index of refraction play into signal transmission, we could calculate the fiber's index of refraction successfully, exemplifying effective physics problem-solving techniques.