# 24.4 Energy in electromagnetic waves  (Page 4/6)

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Integrated Concepts

Police radar determines the speed of motor vehicles using the same Doppler-shift technique employed for ultrasound in medical diagnostics. Beats are produced by mixing the double Doppler-shifted echo with the original frequency. If $1\text{.}\text{50}×{\text{10}}^{9}\text{-Hz}$ microwaves are used and a beat frequency of 150 Hz is produced, what is the speed of the vehicle? (Assume the same Doppler-shift formulas are valid with the speed of sound replaced by the speed of light.)

Integrated Concepts

Assume the mostly infrared radiation from a heat lamp acts like a continuous wave with wavelength $1.50\phantom{\rule{0.25em}{0ex}}\mu \text{m}$ . (a) If the lamp’s 200-W output is focused on a person’s shoulder, over a circular area 25.0 cm in diameter, what is the intensity in ${\text{W/m}}^{2}$ ? (b) What is the peak electric field strength? (c) Find the peak magnetic field strength. (d) How long will it take to increase the temperature of the 4.00-kg shoulder by $2.00º C$ , assuming no other heat transfer and given that its specific heat is $3\text{.}\text{47}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{J/kg}\cdot \text{ºC}$ ?

(a) $4\text{.}\text{07 k}{\text{W/m}}^{2}$

(b) 1.75 kV/m

(c) $5\text{.}\text{84}\phantom{\rule{0.25em}{0ex}}\mu \text{T}$

(d) 2 min 19 s

Integrated Concepts

On its highest power setting, a microwave oven increases the temperature of 0.400 kg of spaghetti by $\text{45}\text{.}0º\text{C}$ in 120 s. (a) What was the rate of power absorption by the spaghetti, given that its specific heat is $3\text{.}\text{76}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{J/kg}\cdot \text{ºC}$ ? (b) Find the average intensity of the microwaves, given that they are absorbed over a circular area 20.0 cm in diameter. (c) What is the peak electric field strength of the microwave? (d) What is its peak magnetic field strength?

Integrated Concepts

Electromagnetic radiation from a 5.00-mW laser is concentrated on a $1\text{.}\text{00}{\text{-mm}}^{2}$ area. (a) What is the intensity in ${\text{W/m}}^{2}$ ? (b) Suppose a 2.00-nC static charge is in the beam. What is the maximum electric force it experiences? (c) If the static charge moves at 400 m/s, what maximum magnetic force can it feel?

(a) $5\text{.}\text{00}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{W/m}}^{2}$

(b) $3\text{.}\text{88}×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{N}$

(c) $5\text{.}\text{18}×{\text{10}}^{-\text{12}}\phantom{\rule{0.25em}{0ex}}\text{N}$

Integrated Concepts

A 200-turn flat coil of wire 30.0 cm in diameter acts as an antenna for FM radio at a frequency of 100 MHz. The magnetic field of the incoming electromagnetic wave is perpendicular to the coil and has a maximum strength of $1\text{.}\text{00}×{\text{10}}^{-\text{12}}\phantom{\rule{0.25em}{0ex}}\text{T}$ . (a) What power is incident on the coil? (b) What average emf is induced in the coil over one-fourth of a cycle? (c) If the radio receiver has an inductance of $2\text{.}\text{50}\phantom{\rule{0.25em}{0ex}}\mu H$ , what capacitance must it have to resonate at 100 MHz?

Integrated Concepts

If electric and magnetic field strengths vary sinusoidally in time, being zero at $t=0$ , then $E={E}_{0}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}2\pi \text{ft}$ and $B={B}_{0}\phantom{\rule{0.25em}{0ex}}\text{sin}\phantom{\rule{0.25em}{0ex}}2\pi \text{ft}$ . Let $f=1\text{.}\text{00 GHz}$ here. (a) When are the field strengths first zero? (b) When do they reach their most negative value? (c) How much time is needed for them to complete one cycle?

(a) $t=0$

(b) $7\text{.}\text{50}×{\text{10}}^{-\text{10}}\phantom{\rule{0.25em}{0ex}}\text{s}$

(c) $1\text{.}\text{00}×{\text{10}}^{-9}\phantom{\rule{0.25em}{0ex}}\text{s}$

Unreasonable Results

A researcher measures the wavelength of a 1.20-GHz electromagnetic wave to be 0.500 m. (a) Calculate the speed at which this wave propagates. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Unreasonable Results

The peak magnetic field strength in a residential microwave oven is $9\text{.}\text{20}×{\text{10}}^{-5}\phantom{\rule{0.25em}{0ex}}\text{T}$ . (a) What is the intensity of the microwave? (b) What is unreasonable about this result? (c) What is wrong about the premise?

(a) $1\text{.}\text{01}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}{\text{W/m}}^{2}$

(b) Much too great for an oven.

(c) The assumed magnetic field is unreasonably large.

Unreasonable Results

An $\text{LC}$ circuit containing a 2.00-H inductor oscillates at such a frequency that it radiates at a 1.00-m wavelength. (a) What is the capacitance of the circuit? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

Unreasonable Results

An $\text{LC}$ circuit containing a 1.00-pF capacitor oscillates at such a frequency that it radiates at a 300-nm wavelength. (a) What is the inductance of the circuit? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) $2\text{.}\text{53}×{\text{10}}^{-\text{20}}\phantom{\rule{0.25em}{0ex}}H$

(b) L is much too small.

(c) The wavelength is unreasonably small.

Consider electromagnetic fields produced by high voltage power lines. Construct a problem in which you calculate the intensity of this electromagnetic radiation in ${\text{W/m}}^{2}$ based on the measured magnetic field strength of the radiation in a home near the power lines. Assume these magnetic field strengths are known to average less than a $\mu T$ . The intensity is small enough that it is difficult to imagine mechanisms for biological damage due to it. Discuss how much energy may be radiating from a section of power line several hundred meters long and compare this to the power likely to be carried by the lines. An idea of how much power this is can be obtained by calculating the approximate current responsible for $\mu T$ fields at distances of tens of meters.

Consider the most recent generation of residential satellite dishes that are a little less than half a meter in diameter. Construct a problem in which you calculate the power received by the dish and the maximum electric field strength of the microwave signals for a single channel received by the dish. Among the things to be considered are the power broadcast by the satellite and the area over which the power is spread, as well as the area of the receiving dish.

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