Question

Alice made a telephone call from her home telephone in New York to her fiancé stationed in Baghdad, about \(10,000 \mathrm{~km}\) away, and the signal was carried on a telephone cable. The following day, Alice called her fiancé again from work using her cell phone, and the signal was transmitted via a satellite \(36,000 \mathrm{~km}\) above the Earth's surface, halfway between New York and Baghdad. Estimate the time taken for the signals sent by (a) the telephone cable and (b) via the satellite to reach Baghdad, assuming that the signal speed in both cases is the same as speed of light, \(c .\) Would there be a noticeable delay in either case?

Short answer

Answer: The time taken for the signal to travel through the telephone cable is estimated to be around 0.0333 seconds, and via the satellite is approximately 0.12 seconds. The difference is about 0.0867 seconds, which is less than a tenth of a second. While this delay might be significant for critical applications such as video conferences or high-speed trading systems, it will not be noticeable during casual conversations.

Step-by-step solution

Given Values and Formula

In the problem, we are given the following values and information: - Distance traveled through telephone cable: 10,000 km - Distance traveled via satellite: 36,000 km - Signal speed: speed of light (c ≈ 3 × 10^8 m/s) To calculate the time taken for the signals, we will use the formula: time (t) = distance (d) / speed (c) Note: We'll need to convert the given distances from kilometers to meters (1 km = 1000 m) before using the formula.

Calculate Time for Telephone Cable

First, we will calculate the time taken for the signal to travel through the telephone cable. Distance (d) = 10,000 km = 10,000 × 1000 m = 10^7 m Speed (c) = 3 × 10^8 m/s Using the formula, we have: t₁ = d₁ / c = (10^7 m) / (3 × 10^8 m/s) = (10^7 / 3 × 10^8) s ≈ 3.33 × 10^(-2) s So, the time taken for the signal to travel through the telephone cable is approximately 0.0333 seconds.

Calculate Time for Satellite Signal

Now, we will calculate the time taken for the signal to travel via the satellite. Distance (d) = 36,000 km = 36,000 × 1000 m = 3.6 × 10^7 m Speed (c) = 3 × 10^8 m/s Using the formula, we have: t₂ = d₂ / c = (3.6 × 10^7 m) / (3 × 10^8 m/s) = (3.6 / 3) × 10^(-1) s = 1.2 × 10^(-1) s So, the time taken for the signal to travel via the satellite is approximately 0.12 seconds.

Compare Times and Determine Noticeable Delay

Now that we have calculated the time taken for the signal to travel through the telephone cable and via the satellite, we can compare them: - Telephone cable: 0.0333 seconds - Satellite: 0.12 seconds The difference between these times is approximately 0.0867 seconds (0.12 - 0.0333 ≈ 0.0867). Since this difference is very small (less than a tenth of a second), it would be difficult for someone to notice the delay between the two modes of communication. However, in some critical applications, such as video conferences or high-speed trading systems, this difference might be significant. For casual conversations, this delay would not be noticeable.

Key concepts

Speed of Light

The speed of light, denoted by the symbol c, is a fundamental physical constant describing the speed at which light travels in a vacuum. It is approximately 3 \times 10^8 meters per second, and it serves as the ultimate speed limit for all forms of energy, matter, and information in the universe.

In the context of telecommunications, the speed of light is crucial since it determines how fast a signal can be transmitted over a given distance. It's important to note that in mediums other than vacuum, such as optical fibers or the Earth's atmosphere, the speed at which light signals travel is slightly less due to the refractive index of that medium.

To calculate the time it takes for a light signal to travel a certain distance, you can use the formula:
\[ t = \frac{d}{c} \]
where t is the time taken, d is the distance the light signal travels, and c is the speed of light. This calculation is essential in understanding signal transmission times in both fiber optics and satellite communications.

Signal Delay

Signal delay, also known as latency, refers to the amount of time it takes for a piece of information, or a signal, to travel from one point to another. In telecommunications, this includes the time from when a message is sent to when it is received.

Signal delay is influenced by various factors including the medium through which the signal travels, the distance between the sender and receiver, and the speed of signal transmission. In the context of the exercise question, the time taken for a signal to travel through a telephone cable versus via a satellite can result in different amounts of signal delay.

Although these signals travel at the speed of light, the delays experienced can differ. This is because the satellite's geostationary orbit significantly increases the transmission path compared to a direct cable. In applications where timing is critical, such as high-frequency trading or real-time gaming, even small delays can have an impact. However, for most everyday uses, such as telephone calls or streaming, these small delays are generally imperceptible to humans.

Telecommunication

Telecommunication involves the transmission of signals over a distance for the purpose of communication. Historically, this has involved the use of optical signals, radio waves, and more recently, digital data transmitted through various mediums.

In today's world, telecommunication is achieved using a variety of methods including wired networks (like telephone cables) and wireless systems (such as satellite communications). The choice between using a cable and a satellite can depend on factors like the required communication distance, the geographical landscape, and the need for mobility.

For example, a telephone cable, typically made of optical fiber, provides a direct and grounded route for signals and often results in lower latency compared to satellites. Satellites, on the other hand, provide wireless communication that can cover a much larger area and are essential for intercontinental communication, broadcasting, and in areas where laying cables is not practical.

The exercise comparison between a telephone cable and a satellite-based call showcases the practical relevance of telecommunication principles. It highlights the engineering trade-offs that have to be considered, such as the balance between signal delay and the scope of coverage.