Chapter 24

(a) Neil A. Armstrong was the first person to walk on the moon. The distance between the earth and the moon is \(3.85 \times 10^{8} \mathrm{~m}\). Find the time it took for his voice to reach earth via radio waves. (b) Someday a person will walk on Mars, which is \(5.6 \times 10^{10} \mathrm{~m}\) from earth at the point of closest approach. Determine the minimum time that will be required for that person's voice to reach earth.

During a flare-up from a sunspot, X-rays (electromagnetic waves) are emitted. If the distance between the sun and the earth is \(1.50 \times 10^{11} \mathrm{~m},\) how long (in minutes) does it take for the X-rays to reach the earth?

In astronomy, distances are often expressed in light-years. One light-year is the distance traveled by light in one year. The distance to Alpha Centauri, the closest star other than our own sun that can be seen by the naked eye, is 4.3 light-years. Express this distance in meters.

Equation \(16.3, y=A \sin (2 \pi f t-2 \pi x / \lambda),\) gives the mathematical representation of a wave oscillating in the \(y\) direction and traveling in the positive \(x\) direction. Let \(y\) in this equation equal the electric field of an electromagnetic wave traveling in a vacuum. The maximum electric field is \(A=156 \mathrm{~N} / \mathrm{C},\) and the frequency is \(f=1.50 \times 10^{8} \mathrm{~Hz} .\) Plot a graph of the electric field strength versus position, using for \(x\) the following values: 0 , \(0.50,1.00,1.50,\) and \(2.00 \mathrm{~m} .\) Plot this graph for \((\) a) a time \(t=0 \mathrm{~s}\) and \((\mathrm{b})\) a time \(t\) that is one-fourth of the wave's period.

A flat coil of wire is used with an LC-tuned circuit as a receiving antenna. The coil has a radius of \(0.25 \mathrm{~m}\) and consists of 450 turns. The transmitted radio wave has a frequency of \(1.2 \mathrm{MHz}\). The magnetic field of the wave is parallel to the normal to the coil and has a maximum value of \(2.0 \times 10^{-13} \mathrm{~T}\). Using Faraday's law of electromagnetic induction and the fact that the magnetic field changes from zero to its maximum value in one-quarter of a wave period, find the magnitude of the average emf induced in the antenna during this time.

A truck driver is broadcasting at a frequency of \(26.965 \mathrm{MHz}\) with a CB (citizen's band) radio. Determine the wavelength of the electromagnetic wave being used.

Obtain the wavelengths in vacuum for (a) blue light whose frequency is \(6.34 \times 10^{14} \mathrm{~Hz}\) and (b) orange light whose frequency is \(4.95 \times 10^{14} \mathrm{~Hz}\). Express your answers in nanometers \(\left(1 \mathrm{nm}=10^{-9} \mathrm{~m}\right)\)

At one time television sets used "rabbit-ears" antennas. Such an antenna consists of a pair of metal rods. The length of each rod can be adjusted to be one-quarter of a wavelength of an electromagnetic wave whose frequency is \(60.0 \mathrm{MHz}\). How long is each rod?

Magnetic resonance imaging, or MRI (see Section 21.7 ), and positron emission tomography, or PET scanning (see Section 32.6), are two medical diagnostic techniques. Both employ electromagnetic waves. For these waves, find the ratio of the MRI wavelength (frequency \(=6.38 \times 10^{7} \mathrm{~Hz}\) ) to the PET scanning wavelength \(\left(\right.\) frequency \(\left.=1.23 \times 10^{20} \mathrm{~Hz}\right)\)

The human eye is most sensitive to light having a frequency of about \(5.5 \times 10^{14} \mathrm{~Hz}\), which is in the yellow-green region of the electromagnetic spectrum. How many wavelengths of this light can fit across the width of your thumb, a distance of about \(2.0 \mathrm{~cm} ?\)

Two radio waves are used in the operation of a cellular telephone. To receive a call, the phone detects the wave emitted at one frequency by the transmitter station or base unit. To send your message to the base unit, your phone emits its own wave at a different frequency. The difference between these two frequencies is fixed for all channels of cell phone operation. Suppose the wavelength of the wave emitted by the base unit is \(0.34339 \mathrm{~m}\) and the wavelength of the wave emitted by the phone is \(0.36205 \mathrm{~m}\). Using a value of \(2.9979 \times 10^{8} \mathrm{~m} / \mathrm{s}\) for the speed of light, determine the difference between the two frequencies used in the operation of a cell phone.

When we look at the star Polaris (the North Star), we are seeing it as it was 680 years ago. How far away from us (in meters) is Polaris?

Two astronauts are \(1.5 \mathrm{~m}\) apart in their spaceship. One speaks to the other. The conversation is transmitted to earth via electromagnetic waves. The time it takes for sound waves to travel at \(343 \mathrm{~m} / \mathrm{s}\) through the air between the astronauts equals the time it takes for the electromagnetic waves to travel to the earth. How far away from the earth is the spaceship?

A lidar (laser radar) gun is an alternative to the standard radar gun that uses the Doppler effect to catch speeders. A lidar gun uses an in frared laser and emits a precisely timed series of pulses of infrared electromagnetic waves. The time for each pulse to travel to the speeding vehicle and return to the gun is measured. In one situation a lidar gun in a stationary police car observes a difference of \(1.27 \times 10^{-7} \mathrm{~s}\) in round-trip travel times for two pulses that are emitted \(0.450 \mathrm{~s}\) apart. Assuming that the speeding vehicle is approaching the police car essentially head-on, determine the speed of the vehicle.

Michelson's setup for measuring the speed of light with the mirrors placed on Mt. San Antonio and Mt. Wilson in California, which are \(35 \mathrm{~km}\) apart. Using a value of \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\) for the speed of light, find the minimum angular speed (in rev \(/ \mathrm{s}\) ) for the rotating mirror.

A communications satellite is in a synchronous orbit that is \(3.6 \times 10^{7} \mathrm{~m}\) directly above the equator. The satellite is located midway between Quito, Equador, and Belém, Brazil, two cities almost on the equator that are separated by a distance of \(3.5 \times 10^{6} \mathrm{~m}\). Find the time it takes for a telephone call to go by way of satellite between these cities. Ignore the curvature of the earth.

A celebrity holds a press conference, which is televised live. A television viewer hears the sound picked up by a microphone directly in front of the celebrity. This viewer is seated \(2.3 \mathrm{~m}\) from the television set. A reporter at the press conference is located \(4.1 \mathrm{~m}\) from the microphone and hears the words directly at the very same instant that the television viewer hears them. Using a value of \(343 \mathrm{~m} / \mathrm{s}\) for the speed of sound, determine the maximum distance between the television viewer and the celebrity.

A laser emits a narrow beam of light. The radius of the beam is \(1.0 \times 10^{-3} \mathrm{~m}\), and the power is \(1.2 \times 10^{-3} \mathrm{~W}\). What is the intensity of the laser beam?

An industrial laser is used to burn a hole through a piece of metal. The average intensity of the light is \(\bar{S}=1.23 \times 10^{9} \mathrm{~W} / \mathrm{m}^{2}\). What is the rms value of (a) the electric field and (b) the magnetic field in the electromagnetic wave emitted by the laser?

The microwave radiation left over from the Big Bang explosion of the universe has an average energy density of \(4 \times 10^{-14} \mathrm{~J} / \mathrm{m}^{3}\). What is the rms value of the electric field of this radiation?

A neodymium-glass laser emits short pulses of high-intensity electromagnetic waves. The electric field of such a wave has an rms value of \(E_{\mathrm{rms}}=2.0 \times 10^{9} \mathrm{~N} / \mathrm{C} .\) Find the average power of each pulse that passes through a \(1.6 \times 10^{-5}-\mathrm{m}^{2}\) surface that is perpendicular to the laser beam.

The average intensity of sunlight at the top of the earth's atmosphere is \(1390 \mathrm{~W} / \mathrm{m}^{2}\). What is the maximum energy that a \(25-\mathrm{m} \times 45\) -m solar panel could collect in one hour in this sunlight?

The mean distance between earth and the sun is \(1.50 \times 10^{11} \mathrm{~m} .\) The average intensity of solar radiation incident on the upper atmosphere of the earth is \(1390 \mathrm{~W} / \mathrm{m}^{2}\). Assuming the sun emits radiation uniformly in all directions, determine the total power radiated by the sun.

An argon-ion laser produces a cylindrical beam of light whose average power is \(0.750 \mathrm{~W}\). How much energy is contained in a 2.50 -m length of the beam?

What fraction of the power radiated by the sun is intercepted by the planet Mercury? The radius of Mercury is \(2.44 \times 10^{6} \mathrm{~m}\), and its mean distance from the sun is \(5.79 \times 10^{10} \mathrm{~m}\). Assume that the sun radiates uniformly in all directions.

The average intensity of sunlight reaching the earth is \(1390 \mathrm{~W} / \mathrm{m}^{2}\). A charge of \(2.6 \times 10^{-8} \mathrm{C}\) is placed in the path of this electromagnetic wave. (a) What is the magnitude of the maximum electric force that the charge experiences? (b) If the charge is moving at a speed of \(3.7 \times 10^{4} \mathrm{~m} / \mathrm{s}\), what is the magnitude of the maximum magnetic force that the charge could experience?

Review Interactive Solution \(\underline{24.31}\) at to see one model for solving this problem. A distant galaxy emits light that has a wavelength of \(434.1 \mathrm{~nm} .\) On earth, the wavelength of this light is measured to be \(438.6 \mathrm{~nm}\). (a) Decide whether this galaxy is approaching or receding from the earth. Give your reasoning. (b) Find the speed of the galaxy relative to the earth.

A distant galaxy emits light that has a wavelength of \(434.1 \mathrm{nm}\). On earth, the wavelength of this light is measured to be \(438.6 \mathrm{nm}\). (a) Decide whether this galaxy is approaching or receding from the earth. Give your reasoning. (b) Find the speed of the galaxy relative to the earth.

Multiple-Concept Example 6 reviews the concepts that play a role in this problem. A speeder is pulling directly away and increasing his distance from a police car that is moving at \(25 \mathrm{~m} / \mathrm{s}\) with respect to the ground. The radar gun in the police car emits an electromagnetic wave with a frequency of \(7.0 \times 10^{9} \mathrm{~Hz}\). The wave reflects from the speeder's car and returns to the police car, where its frequency is measured to be \(320 \mathrm{~Hz}\) less than the emitted frequency. Find the speeder's speed with respect to the ground.

A speeder is pulling directly away and increasing his distance from a police car that is moving at \(25 \mathrm{~m} / \mathrm{s}\) with respect to the ground. The radar gun in the police car emits an electromagnetic wave with a frequency of \(7.0 \times 10^{9} \mathrm{~Hz}\). The wave reflects from the speeder's car and returns to the police car, where its frequency is measured to be \(320 \mathrm{~Hz}\) less than the emitted frequency. Find the speeder's speed with respect to the ground.

Multiple-Concept Example 6 reviews the concepts that play a role in this problem. A speeder is pulling directly away and increasing his distance from a police car that is moving at \(25 \mathrm{~m} / \mathrm{s}\) with respect to the ground. The radar gun in the police car emits an electromagnetic wave with a frequency of \(7.0 \times 10^{9} \mathrm{~Hz}\). The wave reflects from the speeder's car and returns to the police car, where its frequency is measured to be \(320 \mathrm{~Hz}\) less than the emitted frequency. Find the speeder's speed with respect to the ground.

A distant galaxy is simultaneously rotating and receding from the earth. As the drawing shows, the galactic center is receding from the earth at a relative speed of \(u_{\mathrm{G}}=1.6 \times 10^{6} \mathrm{~m} / \mathrm{s} .\) Relative to the center, the tangential speed is \(v_{\mathrm{T}}=0.4 \times 10^{6} \mathrm{~m} / \mathrm{s}\) for locations \(A\) and \(B\), which are equidistant from the center. When the frequencies of the light coming from regions \(A\) and \(B\) are measured on earth, they are not the same and each is different from the emitted frequency of \(6.200 \times 10^{14} \mathrm{~Hz}\). Find the measured frequency for the light from (a) region \(A\) and (b) region \(B\).

Linearly polarized light is incident on a piece of polarizing material. What is the ratio of the transmitted light intensity to the incident light intensity when the angle between the transmission axis and the incident electric field is (a) \(25^{\circ}\) and (b) \(65^{\circ}\) ?

For each of the three sheets of polarizing material shown in the drawing, the orientation of the transmission axis is labeled relative to the vertical. The incident beam of light is unpolarized and has an intensity of \(1260.0 \mathrm{~W} / \mathrm{m}^{2}\). What is the intensity of the beam transmitted through the three sheets when \(\theta_{1}=19.0^{\circ}, \theta_{2}=55.0^{\circ},\) and \(\theta_{3}=100.0^{\circ} ?\)

Suppose that the transmission axis of the first analyzer is rotated \(27^{\circ}\) relative to the transmission axis of the polarizer, and that the transmission axis of each additional analyzer is rotated \(27^{\circ}\) relative to the transmission axis of the previous one. What is the minimum number of analyzers needed for the light reaching the photocell to have an intensity that is reduced by at least a factor of 100 relative to that striking the first analyzer?

Some of the X-rays produced in an X-ray machine have a wavelength of \(2.1 \mathrm{nm}\). What is the frequency of these electromagnetic waves?

The maximum strength of the magnetic field in an electromagnetic wave is \(3.3 \times 10^{-6} \mathrm{~T}\). What is the maximum strength of the wave's electric field?

TV channel 3 (VHF) broadcasts at a frequency of 63.0 MHz. TV channel 23 (UHF) broadcasts at a frequency of \(527 \mathrm{MHz}\). Find the ratio (VHF/UHF) of the wavelengths for these channels.

A future space station in orbit about the earth is being powered by an electromagnetic beam from the earth. The beam has a cross-sectional area of \(135 \mathrm{~m}^{2}\) and transmits an average power of \(1.20 \times 10^{4} \mathrm{~W}\). What are the rms values of the (a) electric and (b) magnetic fields?

Suppose that the police car in that example is moving to the right at \(27 \mathrm{~m} / \mathrm{s},\) while the speeder is coming up from behind at a speed of \(39 \mathrm{~m} / \mathrm{s}\), both speeds being with respect to the ground. Assume that the electromagnetic wave emitted by the radar gun has a frequency of \(8.0 \times 10^{9} \mathrm{~Hz}\). Find the difference between the frequency of the wave that returns to the police car after reflecting from the speeder's car and the original frequency emitted by the police car.

The electromagnetic wave that delivers a cellular phone call to a car has a magnetic field with an rms value of \(1.5 \times 10^{-10} \mathrm{~T}\). The wave passes perpendicularly through an open window, the area of which is \(0.20 \mathrm{~m}^{2}\). How much energy does this wave carry through the window during a 45 -s phone call?

In a traveling electromagnetic wave, the electric field is represented mathematically as $$ E=E_{0} \sin \left[\left(1.5 \times 10^{10} \mathrm{~s}^{-1}\right) t-\left(5.0 \times 10^{1} \mathrm{~m}^{-1}\right) x\right] $$ where \(E_{0}\) is the maximum field strength. (a) What is the frequency of the wave? (b) This wave and the wave that results from its reflection can form a standing wave, in a way similar to that in which standing waves can arise on a string (see Section 17.5). What is the separation between adjacent nodes in the standing wave?

A beam of polarized light has an average intensity of \(15 \mathrm{~W} / \mathrm{m}^{2}\) and is sent through a polarizer. The transmission axis makes an angle of \(25^{\circ}\) with respect to the direction of polarization. Determine the rms value of the electric field of the transmitted beam.

\(E_{\mathrm{rms}}=2800 \mathrm{~N} / \mathrm{C}\). (a) What is the average intensity of the radiation? (b) The radiation is focused on a person's leg over a circular area of radius \(4.0 \mathrm{~cm} .\) What is the average power delivered to the leg? (c) The portion of the leg being radiated has a mass of \(0.28 \mathrm{~kg}\) and a specific heat capacity of \(3500 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right)\). How long does it take to raise its temperature by \(2.0 \mathrm{C}^{\circ} ?\) Assume that there is no other heat transfer into or out of the portion of the leg being heated.

A heat lamp emits infrared radiation whose rms electric field is \(E_{\mathrm{rms}}=2800 \mathrm{~N} / \mathrm{C} .\) (a) What is the average intensity of the radiation? (b) The radiation is focused on a person's leg over a circular area of radius \(4.0 \mathrm{~cm}\). What is the average power delivered to the leg? (c) The portion of the leg being radiated has a mass of \(0.28 \mathrm{~kg}\) and a specific heat capacity of \(3500 \mathrm{~J} /\left(\mathrm{kg} \cdot \mathrm{C}^{\circ}\right) .\) How long does it take to raise its temperature by \(2.0 \mathrm{C}^{\circ}\) ? Assume that there is no other heat transfer into or out of the portion of the leg being heated.

Interactive Solution \(\underline{24.53}\) at provides one model for problems like this one. The drawing shows an edge-on view of the solar panels on a communications satellite. The dashed line specifies the normal to the panels. Sunlight strikes the panels at an angle \(\theta\) with respect to the normal. If the solar power impinging on the panels is \(2600 \mathrm{~W}\) when \(\theta=65^{\circ}\), what is it when \(\theta=25^{\circ} ?\)

A certain type of laser emits light of known frequency. The light, however, occurs as a series of short pulses, each lasting for a time \(t_{0}\). (a) How is the wavelength of the light related to its frequency? (b) How is the length (in meters) of each pulse related to the time \(t_{0}\) ?

(a) Suppose that the magnitude \(E\) of the electric field in an electromagnetic wave triples. By what factor does the intensity \(S\) of the wave change? (b) The magnitude \(B\) of the magnetic field is much smaller than \(E\) because, according to Equation 24.3, \(B=E / c\), where \(c\) is the speed of light in a vacuum. If \(B\) triples, by what factor does the intensity change? Account for your answers.

A source is radiating light waves uniformly in all directions. At a certain distance \(r\) from the source a person measures the average intensity of the waves. (a) Does the average intensity increase, decrease, or remain the same, as \(r\) increases? (b) If the magnitude of the electric field is determined from the average intensity, is the electric field the rms value or the peak value? In both cases, justify your answers. A light bulb emits light uniformly in all directions. The average emitted power is \(150.0 \mathrm{~W}\). At a distance of \(5.00 \mathrm{~m}\) from the bulb, determine (a) the average intensity of the light, (b) the rms value of the electric field, and (c) the peak value of the electric field.

An electric charge is placed in a laser beam. Does a stationary charge experience a force due to (a) the electric field and (b) the magnetic field of the electromagnetic wave? Now suppose that the charge is moving perpendicular to the magnetic field of the beam. Does it experience (c) an electric force and (d) a magnetic force? Account for your answers.