Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 31 - Maxwell's Equations and Electromagnetic Waves

Problem 54

Chapter 30, Problem 54

Radio-controlled clocks throughout the United States receive a radio signal from a transmitter in Fort Collins, Colorado, that accurately (within a microsecond) marks the beginning of each minute. A slight amount of time is added by a clock at any location to correct for the signal travel time to the clock from Fort Collins. Assuming Fort Collins is no more than 3000 km from any point in the U.S., what is the longest travel-time delay?

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Determine the speed of the radio signal. Since radio signals are electromagnetic waves, they travel at the speed of light in a vacuum, which is approximately 3 × 10⁸ m/s.

Convert the maximum distance from Fort Collins to any point in the U.S. into meters. Given that the maximum distance is 3000 km, convert it to meters using the conversion factor: 1 km = 1000 m. Thus, 3000 km = 3000 × 1000 = 3 × 10⁶ m.

Use the formula for time delay:

t = \frac{d}{v}

, where

t

is the travel time,

d

is the distance, and

v

is the speed of the signal. Substitute

d =3×10⁶

m and

v =3×10⁸

m/s into the formula.

Simplify the expression to calculate the time delay:

t = \frac{3×10⁶}{3×10⁸}

. This will give the time delay in seconds.

Interpret the result. The calculated time delay represents the longest travel-time delay for the radio signal to reach any point in the U.S. from Fort Collins, Colorado.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Speed of Light

The speed of light in a vacuum is approximately 299,792 kilometers per second. This fundamental constant is crucial for understanding how quickly signals, such as radio waves, travel. Since radio signals travel at the speed of light, knowing this speed allows us to calculate the time it takes for a signal to reach a receiver from a transmitter.

Time Delay Calculation

Time delay refers to the duration it takes for a signal to travel from one point to another. To calculate the time delay for a radio signal traveling a distance of 3000 km, one can use the formula: time = distance/speed. This calculation is essential for determining how much time should be added to a clock to account for the signal's travel time.

Signal Propagation

Signal propagation is the way in which radio waves travel through space. Factors such as distance, medium, and environmental conditions can affect the speed and quality of the signal. Understanding signal propagation is important for accurately determining the effective range and timing of radio-controlled devices, such as clocks.

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