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Ch 16: Traveling Waves
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 16: Traveling WavesProblem 84
Chapter 16, Problem 84

A communications truck with a 44-cm-diameter dish receiver on the roof starts out 10 km from its base station. It drives directly away from the base station at 50 km/h for 1.0 h, keeping the receiver pointed at the base station. The base station antenna broadcasts continuously with 2.5 kW of power, radiated uniformly in all directions. How much electromagnetic energy does the truck's dish receive during that 1.0 h?

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Determine the total distance the truck travels during the 1.0 h. Since the truck is moving at a constant speed of 50 km/h, the total distance traveled is given by the formula: \( \text{distance} = \text{speed} \times \text{time} \). Add this distance to the initial distance of 10 km to find the final distance from the base station.
Calculate the power per unit area (intensity) of the electromagnetic wave at the truck's final distance. The intensity of the wave at a distance \( r \) from the source is given by: \( I = \frac{P}{4 \pi r^2} \), where \( P \) is the total power radiated by the base station (2.5 kW) and \( r \) is the distance from the source.
Determine the area of the dish receiver on the truck. The dish is circular with a diameter of 44 cm, so its radius is \( r = \frac{44}{2} \) cm = 22 cm = 0.22 m. The area of the dish is given by: \( A = \pi r^2 \).
Calculate the power received by the dish. The power received is the product of the intensity of the wave at the truck's location and the area of the dish: \( P_{\text{received}} = I \times A \).
Determine the total electromagnetic energy received by the dish during the 1.0 h. Energy is the product of power and time: \( E = P_{\text{received}} \times t \), where \( t \) is the time in seconds (1.0 h = 3600 s).

Key Concepts

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

Electromagnetic Radiation

Electromagnetic radiation refers to the waves of the electromagnetic field, propagating through space, carrying energy. It includes a range of wavelengths, from radio waves to gamma rays. In this scenario, the base station broadcasts radio waves, which are a form of electromagnetic radiation, and the dish receiver captures this energy.
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Inverse Square Law

The inverse square law states that the intensity of radiation from a point source decreases with the square of the distance from the source. This means that as the truck moves away from the base station, the power received by the dish diminishes significantly due to the increasing distance, affecting the total energy captured over time.
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Power and Energy Relationship

Power is the rate at which energy is transferred or converted, measured in watts (W). In this context, the base station transmits at 2.5 kW, meaning it emits 2.5 kilojoules of energy per second. To find the total energy received by the dish over 1 hour, one must multiply the power by the time duration, taking into account the distance and the inverse square law.
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Related Practice
Textbook Question

An avant-garde composer wants to use the Doppler effect in his new opera. As the soprano sings, he wants a large bat to fly toward her from the back of the stage. The bat will be outfitted with a microphone to pick up the singer's voice and a loudspeaker to rebroadcast the sound toward the audience. The composer wants the sound the audience hears from the bat to be, in musical terms, one half-step higher in frequency than the note they are hearing from the singer. Two notes a half-step apart have a frequency ratio of 21/12 = 1.059. With what speed must the bat fly toward the singer?

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Textbook Question

A loudspeaker, mounted on a tall pole, is engineered to emit 75% of its sound energy into the forward hemisphere, 25% toward the back. You measure an 85 dB sound intensity level when standing 3.5 m in front of and 2.5 m below the speaker. What is the speaker's power output?

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Textbook Question

A physics professor demonstrates the Doppler effect by tying a 600 Hz sound generator to a 1.0-m-long rope and whirling it around her head in a horizontal circle at 100 rpm. What are the highest and lowest frequencies heard by a student in the classroom?

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