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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
Chapter 30, Problem 59

The average intensity of a particular TV station’s signal is 1.0 x 10-13 W/m2 when it arrives at a 33-cm-diameter satellite TV antenna. (a) Calculate the total energy received by the antenna during 3.0 hours of viewing this station’s programs. (b) Estimate the amplitudes of the E and B fields of the EM wave.

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Step 1: Calculate the area of the satellite TV antenna. The antenna is circular, so its area can be calculated using the formula for the area of a circle: A = πr². The radius (r) is half the diameter, so r = 33 cm / 2 = 0.165 m. Substitute this value into the formula to find the area.
Step 2: Determine the total energy received by the antenna during 3.0 hours. The power received by the antenna is given by P = I × A, where I is the intensity of the signal (1.0 × 10⁻¹³ W/m²) and A is the area of the antenna. Multiply the power by the total time (3.0 hours converted to seconds: 3.0 × 3600 s) to find the total energy received: E = P × t.
Step 3: Use the relationship between intensity and the electric field amplitude (E₀) to estimate the amplitude of the electric field. The intensity of an electromagnetic wave is related to the electric field amplitude by the formula I = (1/2)ε₀cE₀², where ε₀ is the permittivity of free space (8.85 × 10⁻¹² F/m) and c is the speed of light (3.0 × 10⁸ m/s). Rearrange this formula to solve for E₀: E₀ = √(2I / (ε₀c)).
Step 4: Estimate the amplitude of the magnetic field (B₀) using the relationship between the electric and magnetic fields in an electromagnetic wave. The amplitudes are related by the equation B₀ = E₀ / c, where c is the speed of light. Use the value of E₀ calculated in the previous step to find B₀.
Step 5: Summarize the results. The total energy received by the antenna during 3.0 hours is calculated in step 2. The amplitudes of the electric and magnetic fields (E₀ and B₀) are estimated in steps 3 and 4, respectively. These values provide a complete understanding of the problem.

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Key Concepts

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

Intensity of a Wave

Intensity is defined as the power per unit area carried by a wave, typically measured in watts per square meter (W/m²). It quantifies how much energy is transmitted through a given area and is crucial for understanding how much energy a receiving antenna can capture from an electromagnetic (EM) wave.
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Energy Calculation

To calculate the total energy received by the antenna, one must multiply the intensity of the signal by the area of the antenna and the time duration of exposure. The formula used is Energy = Intensity × Area × Time, which allows for determining the total energy captured over a specified period.
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Electromagnetic Fields

Electromagnetic waves consist of oscillating electric (E) and magnetic (B) fields that are perpendicular to each other and to the direction of wave propagation. The amplitudes of these fields can be estimated from the intensity of the wave, as they are related through the equation I = (1/2) * ε₀ * c * E², where ε₀ is the permittivity of free space and c is the speed of light.
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Related Practice
Textbook Question

A global positioning system (GPS) functions by determining the travel times for EM waves from various satellites to a moving GPS receiver on Earth (car or hiker). If the receiver is to detect a change in the receiver’s position on the order of 3 m, what is the associated change in travel time (in ns) that must be measured?

Textbook Question

Show that displacement current, ε₀ (dΦE/dt), has the SI units of amperes.

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Textbook Question

What length of antenna would be appropriate for a portable device that could receive satellite TV?

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Textbook Question

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?

Textbook Question

Who will hear the voice of a singer first: a person in the balcony 50.0 km away from the stage (see Fig. 31–26), or a person 1800 km away at home whose ear is next to the radio listening to a live broadcast? Roughly how much sooner? Assume the microphone is a few centimeters from the singer and the temperature is 20℃.


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Textbook Question

A powerful laser portrayed in a movie provides a 3-mm-diameter beam of green light with a power of 3 W. A good agent inside a spacecraft aims the laser beam at an enemy astronaut hovering outside. The mass of the enemy astronaut is 120 kg and the spacecraft 185,000 kg. (a) Determine the “radiation-pressure” force exerted on the enemy by the laser beam assuming her suit is perfectly reflecting. (b) If the enemy is 30 m from the spacecraft’s center of mass, estimate the gravitational force the spacecraft exerts on the enemy. (c) Which of the two forces is larger, and by what factor?

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