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1. Intro to Physics Units1h 29m
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17. Periodic Motion2h 9m
18. Waves & Sound3h 40m
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22. The First Law of Thermodynamics1h 26m
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30. Induction and Inductance3h 38m
31. Alternating Current2h 37m
32. Electromagnetic Waves2h 14m
33. Geometric Optics2h 57m
34. Wave Optics1h 15m
35. Special Relativity2h 10m

18. Waves & Sound

Wave Intensity

Physics

18. Waves & Sound

Wave Intensity

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Problem 84

Textbook Question

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?

Verified step by step guidance

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).

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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.

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.

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

A jet plane at takeoff can produce sound of intensity 10.0 W/m² at 30.0 m away. But you prefer the tranquil sound of normal conversation, which is 1.0 μW/m². Assume that the plane behaves like a point source of sound. (a) What is the closest dis-tance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at takeoff?

At a distance of 7.00 x 10¹² m from a star, the intensity of the radiation from the star is 15.4 W/m². Assuming that the star radiates uniformly in all directions, what is the total power output of the star?

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Textbook Question

The intensity of electromagnetic waves from the sun is 1.4 kW/m² just above the earth's atmosphere. Eighty percent of this reaches the surface at noon on a clear summer day. Suppose you think of your back as a 30 cm x 50 cm rectangle. How many joules of solar energy fall on your back as you work on your tan for 1.0 h?

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Textbook Question

LASIK eye surgery uses pulses of laser light to shave off tissue from the cornea, reshaping it. A typical LASIK laser emits a 1.0-mm-diameter laser beam with a wavelength of 193 nm. Each laser pulse lasts 15 ns and contains 1.0 mJ of light energy. During the very brief time of the pulse, what is the intensity of the light wave?

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Textbook Question

A sound wave with intensity 2.0 x 10^-3 W/m² is perceived to be modestly loud. Your eardrum is 6.0 mm in diameter. How much energy will be transferred to your eardrum while listening to this sound for 1.0 min?

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Textbook Question

A concert loudspeaker suspended high above the ground emits 35 W of sound power. A small microphone with a 1.0 cm² area is 50 m from the speaker. What is the sound intensity at the position of the microphone?

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Textbook Question

A sound source is located somewhere along the x-axis. Experiments show that the same wave front simultaneously reaches listeners at x = -7.0 m and x = ＋3.0 m. What is the x-coordinate of the source?

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Multiple Choice

Which of the following changes would result in an increase in wave intensity?

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