Chapter 32

(a) How much time does it take light to travel from the moon to the earth, a distance of \(384,000 \mathrm{km}\) ? (b) Light from the star Sirius takes 8.61 years to reach the earth. What is the distance from earth to Sirius in kilometers?

In a TV picture, ghost images are formed when the signal from the transmitter travels to the receiver both directly and indirectly after reflection from a building or other large metallic mass. In a 25 -inch set, the ghost is about 1.0 \(\mathrm{cm}\) to the right of the principal image if the reflected signal arrives 0.60\(\mu\) s after the principal signal. In this case, what is the difference in path lengths for the two signals?

For an electromagnetic wave propagating in air, determine the frequency of a wave with a wavelength of (a) \(5.0 \mathrm{km} ;\) (b) 5.0 \(\mathrm{m}\) (c) \(5.0 \mu \mathrm{m} ;\) (d) 5.0 \(\mathrm{nm}\)

There are two categories of ultraviolet light. Ultraviolet A (UVA) has a wavelength ranging from 320 \(\mathrm{nm}\) to 400 \(\mathrm{nm}\) . It is not so harmful to the skin and is necessary for the production of vitamin D. UVB, with a wavelength between 280 \(\mathrm{nm}\) and 320 \(\mathrm{mm}\) , is much more dangerous because it causes skin cancer. (a) Find the frequency ranges of UVA and UVB. (b) What are the ranges of the wave numbers for UVA and UVB?

An electromagnetic wave of wavelength \(435 \mathrm{~nm}\) is traveling in vacuum in the \(-z\) -direction. The electric field has amplitude \(2.70 \times 10^{-3} \mathrm{~V} / \mathrm{m}\) and is parallel to the \(x\) -axis. What are (a) the frequency and (b) the magnetic-field amplitude? (c) Write the vector equations for \(\overrightarrow{\boldsymbol{E}}(z, t)\) and \(\overrightarrow{\boldsymbol{B}}(z, t)\).

A sinusoidal electromagnetic wave of frequency \(6.10 \times 10^{14} \mathrm{Hz}\) travels in vacuum in the \(+z\) -direction. The \(\overrightarrow{\boldsymbol{B}}\) -field is parallel to the \(y\) -axis and has amplitude \(5.80 \times 10^{-4}\) T. Write the vector equations for \(\overrightarrow{\boldsymbol{E}}(z, t)\) and \(\overrightarrow{\boldsymbol{B}}(z, t) .\)

The electric field of a sinusoidal electromagnetic wave obeys the equation \(E=-(375 \mathrm{V} / \mathrm{m}) \sin \left[\left(5.97 \times 10^{15} \mathrm{rad} / \mathrm{s}\right) t+\right.\) \(\left(1.99 \times 10^{7} \mathrm{rad} / \mathrm{m}\right) x ]\) . (a) What are the amplitudes of the electric and magnetic fields of this wave? (b) What are the frequency, wavelength, and period of the wave? Is this light visible to humans?(c) What is the speed of the wave?

Radio station WCCO in Minneapolis broadcasts at a frequency of 830 \(\mathrm{kHz}\) . At a point some distance from the transmitter, the magnetic- field amplitude of the electromagnetic wave from \(\mathrm{WCCO}\) is \(4.82 \times 10^{-11}\) T. Calculate (a) the wavelength; (b) the wave number; (c) the angular frequency; (d) the electric-field amplitude.

The electric-field amplitude near a certain radio transmitter is \(3.85 \times 10^{-3} \mathrm{V} / \mathrm{m}\) What is the amplitude of \(\overrightarrow{\boldsymbol{B}} ?\) How does this compare in magnitude with the earth's field?

An electromagnetic wave with frequency \(5.70 \times 10^{14} \mathrm{Hz}\) propagates with a speed of \(2.17 \times 10^{8} \mathrm{m} / \mathrm{s}\) in a certain piece of glass. Find (a) the wavelength of the wave in the glass; (b) the wavelength of a wave of the same frequency propagating in air; (c) the index of refraction \(n\) of the glass for an electromagnetic wave with this frequency; (d) the dielectric constant for glass at this frequency, assuming that the relative permeability is unity.

An electromagnetic wave with frequency 65.0 \(\mathrm{Hz}\) travels in an insulating magnetic material that has dielectric constant 3.64 and relative permeability 5.18 at this frequency. The electric field has amplitude \(7.20 \times 10^{-3} \mathrm{V} / \mathrm{m}\) . (a) What is the speed of propagation of the wave? (b) what is the wavelength of the wave? (c) What is the amplitude of the magnetic field? (d) What is the intensity of the wave?

We can reasonably model a 75-W incandescent light-bulb as a sphere 6.0 \(\mathrm{cm}\) in diameter. Typically, only about 5\(\%\) of the energy goes to visible light; the rest goes largely to nonvisible infrared radiation. (a) What is the visible-light intensity (in \(\mathrm{W} / \mathrm{m}^{2} )\) at the surface of the bulb? (b) What are the amplitudes of the electric and magnetic fields at this surface, for a sinusoidal wave with this intensity?

A sinusoidal electromagnetic wave from a radio station passes perpendicularly through an open window that has area \(0.500 \mathrm{m}^{2} .\) At the window, the electric field of the wave has rms value 0.0200 \(\mathrm{V} / \mathrm{m}\) . How much energy docs this wave carry through the window during a 30.0 -s commercial?

You are a NASA mission specialist on your first flight aboard the space shuttle. Thanks to your extensive training in pliysics, you have been assigned to evaluate the performance of a new radio transmitter on board the International Space Station (ISS). Perched on the shuttle's movable arm, you aim a sensitive detector at the ISS, which is 2.5 \(\mathrm{km}\) away. You find that the electric-field amplitude of the radio waves coming from the ISS transmitter is 0.090 \(\mathrm{V} / \mathrm{m}\) and that the frequency of the waves is 244 \(\mathrm{MHz}\) . Find the following: (a) the intensity of the radio wave at your location; (b) the magnetic-field amplitude of the wave at your location; (c) the total power output of the ISS radio transmitter. (d) What assumptions, if any, did you make in your calculations?

The intensity of a cylindrical laser beam is 0.800 \(\mathrm{W} / \mathrm{m}^{2}\) The cross-sectional area of the beam is \(3.0 \times 10^{-4} \mathrm{m}^{2}\) and the intensity is uniform across the cross section of the beam. (a) What is the average power output of the laser? (b) What is the rms value of the electric field in the beam?

A space probe \(2.0 \times 10^{10} \mathrm{m}\) from a star measures the total intensity of electrumagnetic radiation from the star to be \(5.0 \times 10^{3} \mathrm{W} / \mathrm{m}^{2} .\) If the star radiates uniformly in all directions, what is its total average power output?

A simusoidal electromagnetic wave emitted by a cellular phone has a wavelength of 35.4 \(\mathrm{cm}\) and an electric-field amplitude of \(5.40 \times 10^{-2} \mathrm{V} / \mathrm{m}\) at a distance of 250 \(\mathrm{m}\) from the antenna. Calculate (a) the frequency of the wave; (b) the magnetic-field amplitude; (c) the intensity of the wave.

An intense light source radiates uniformly in all directions. At a distance of 5.0 \(\mathrm{m}\) from the source, the radiation pressure on a perfectly absorbing surface is \(9.0 \times 10^{-6} \mathrm{Pa}\) . What is the total average power output of the source?

Public television station KQED in San Francisco broadcasts a sinusoidal radio signal at a power of 316 \(\mathrm{kW}\) . Assume that the wave spreads out uniformly into a hemisphere above the ground. At a home 5.00 \(\mathrm{km}\) away from the antenna, (a) what average pressure does this wave exert on a totally reflecting surface, \((b)\) what are the amplitudes of the electric and magnetic fields of the wave, and (c) what is the aver- age density of the energy this wave carries? (d) For the energy density in part (c), what percentage is due to the electric field and what percentage is due to the magnetic field?

If the intensity of direct sunlight at a point on the earth's surface is \(0.78 \mathrm{kW} / \mathrm{m}^{2},\) find \((\mathrm{a})\) the average momentum density (momentum per unit volume) in the sunlight and (b) the average momentum flow rate in the sunlight.

In the \(25-\) fil Space Simulator facility at NASA's Jet Propulsion Laboratory, a bank of overhead arc lamps can produce light of intensity 2500 \(\mathrm{W} / \mathrm{m}^{2}\) at the floor of the facility. (This simulates the intensity of sunlight near the planet Venus.) Find the average radiation pressure (in pascals and in atmospheres) on (a) a totally absorbing section of the floor and (b) a totally reflecting section of the floor. (c) Find the average momentum density (momentum per unit volume) in the light at the floor.

An electromagnetic standing wave in air of frequency 750 MHz is set up between two conducting planes 80.0 \(\mathrm{cm}\) apart. At which positions between the planes could a point charge be placed at rest so that it would remain at rest? Explain.

A standing electromagnetic wave in a certain material has frequency \(2.20 \times 10^{10} \mathrm{Hz}\) . The nodal planes of \(\overrightarrow{\boldsymbol{B}}\) are 3.55 \(\mathrm{mm}\) apart. Find (a) the wavelength of the wave in this material; (b) the distance between adjacent nodal planes of the \(\overrightarrow{\boldsymbol{E}}\) field: (c) the speed of propagation of the wave.

An electromagnetic standing wave in air has frequency 75.0 MHz. (a) What is the distance between nodal planes of the \(\overrightarrow{\boldsymbol{E}} \) field? (b) What is the distance between a nodal plane of \(\vec{E}\) and the closest nodal plane of \(\overrightarrow{\boldsymbol{B}} ?\)

An electromagnetic standing wave in a certain material has frequency \(1.20 \times 10^{10} \mathrm{Hz}\) and speed of propagation \(210 \times 10^{8} \mathrm{m} / \mathrm{s}\) . (a) What is the distance between a nodal plane of \(\overrightarrow{\boldsymbol{B}}\) and the closest antinodal plane of \(\overrightarrow{\boldsymbol{B}} ?\) (b) What is the distance between an antinodal plane of \(\overrightarrow{\boldsymbol{E}}\) and the closest antinodal plane of \(\overrightarrow{\boldsymbol{B}}\) ? (c) What is the distance between a nodal plane of \(\overrightarrow{\boldsymbol{E}}\) and the closest nodal plane of \(\overrightarrow{\boldsymbol{B}} ?\)

The microwaves in a certain microwave oven have a wavelength of \(12.2 \mathrm{cm} .\) (a) How wide must this oven be so that it will contain five antinodal planes of the electric field along its width in the standing wave pattern? (b) What is the frequency of these microwaves? (c) Suppose a manufacturing error occurred and the oven was made 5.0 \(\mathrm{cm}\) longer than specificd in part (a). In this case, what would have to be the frequency of the microwaves for there still to be five antinodal planes of the electric field along the width of the oven?

A satellite 575 \(\mathrm{km}\) above the earth's surface transmits sinusoidal electromagnetic waves of frequency 92.4 \(\mathrm{MHz}\) uniformly in all directions, with a power of 25.0 \(\mathrm{kW}\) . (a) What is the intensity of these waves as they reach a receiver at the surface of the earth directly below the satellite? (b) What are the amplitudes of the electric and magnetic fields at the receiver? (c) If the receiver has a totally absorbing panel measuring 15.0 \(\mathrm{cm}\) by 40.0 \(\mathrm{cm}\) oriented with its plane perpendicular to the direction the waves travel, what average force do these waves exert on the panel? Is this force large enough to cause significant effects?

A plane sinusoidal clectromagnetic wave in air has a wavelength of 3.84 \(\mathrm{cm}\) and an \(\overrightarrow{\boldsymbol{E}}\) -field amplitude of 1.35 \(\mathrm{V} / \mathrm{m}\) . (a) What is the frequency? (b) What is the \(\overrightarrow{\boldsymbol{B}}\) -field amplitude? (c) What is the intensity?(d) What average force does this radiation exert on a totally absorbing surface with area 0.240 \(\mathrm{m}^{2}\) perpendicular to the direction of propagation?

A small helium-neon laser emits red visible light with a power of 3.20 \(\mathrm{mW}\) in a beam that has a diameter of 2.50 \(\mathrm{mm}\) . (a) What are the amplitudes of the electric and magnetic fields of the light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a \(1.00-\mathrm{m}\) length of the beam?

The sun emits energy in the form of electromagnetic waves at a rate of \(3.9 \times 10^{26} \mathrm{W}\) . This energy is produced by nuclear reactions deep in the sun's interior. (a) Find the intensity of electromagnetic radiation and the radiation pressure on an absorbing object at the surface of the sun (radius \(r=R=6.96 \times 10^{5} \mathrm{km}\)) and at \(r=R / 2\) , in the sun's interior. Ignore any scattering of the waves as they move radially outward from the center of the sun. Compare to the values given in Section 32.4 for sunlight just before it enters the earth's atmosphere. (b) The gas pressure at the sun's surface is about \(1.0 \times 10^{4} \mathrm{Pa}\) ; at \(r=R / 2,\) the gas pressure is calculated from solar models to be about \(4.7 \times 10^{33} \mathrm{Pa}\) Comparing with your results in part (a), would you expect that radiation pressure is an important factor in determining the structure of the sun? Why or why not?

It has been proposed to place solar-power-collecting satellites in earth orbit. The power they collect would be beamed down to the earth as microwave radiation. For a microwave beam with a cross-sectional area of 36.0 \(\mathrm{m}^{2}\) and a total power of 2.80 \(\mathrm{kW}\) at the earth's surface, what is the amplitude of the electric field of the beam at the earth's surface?

Two square reflectors, each 1.50 \(\mathrm{cm}\) on a side and of mass 4.00 \(\mathrm{g}\) , are located at opposite ends of a thin, extremely light, \(1.00-\mathrm{m}\) rod that can rotate without friction and in a vacuum about an axle perpendicular to it through its center (Fig. 32.24\()\) . These reficctors are small enough to be treated as point masses in moment-of-inertia calculations. Both reflectors are illuminated on one face by a sinusoidal light wave having an electric field of amplitude 1.25 \(\mathrm{N} / \mathrm{C}\) that falls uniformly on both surfaces and always strikes them perpendicular to the plane of their surfaces. One reflector is covered with a perfectly absorbing coating, and the other is covered with a perfectly reflecting coating. What is the angular acceleration of this device?

The plane of a flat surface is perpendicular to the propagation direction of an electromagnetic wave of intensity \(1 .\) The surface absorbs a fraction \(w\) of the incident intensity, where \(0 \leq w \leq 1,\) and reflects the rest. (a) Show that the radiation pressure on the surface equals \((2-w) I / c .\) (b) Show that this expression gives the correct results for a surface that is (i) totally absorbing and (ii) totally reflective. (c) For an incident intensity of \(1.40 \mathrm{kW} / \mathrm{m}^{2},\) what is the radiation pressure for 90\(\%\) absorption? For 90\(\%\) reflection?

A source of sinusoidal electromagnetic waves radiates uniformly in all directions. At 10.0 \(\mathrm{m}\) from this source, the amplitude of the electric field is measured to be 1.50 \(\mathrm{N} / \mathrm{C}\) . What is the electricfield amplitude at a distance of 20.0 \(\mathrm{cm}\) from the source?

A circular loop of wire can be used us a radio antenna. If a 18.0-cm-diameter antenna is located 2.50 \(\mathrm{km}\) from a 95.0 -MHz source with a total power of 55.0 \(\mathrm{kW}\) , what is the maximum emf induced in the loop? (Assume that the plane of the antenna loop is perpendicular to the direction of the radiation's magnetic field and that the source radiates uniformly in all directions.)

In a certain experiment, a radio transmitter emits sinusoidal electromagnetic waves of frequency 110.0 \(\mathrm{MHz}\) in opposite directions inside a narrow cavity with reflectors at both ends, causing a standing wave pattern to occur. (a) How far apart are the nodal planes of the magnetic field? (b) If the standing wave pattern is determined to be in its eighth harmonic, how long is the cavity?

Interplanetary space contains many small particles referred to as interplanetary dust. Radiation pressure from the sun sets a lower limit on the size of such dust particles. To see the origin of this limit, consider a spherical dust particle of radius \(R\) and mass density \(\rho\) (a) Write an expression for the gravitational force exerted on this particle by the sun (mass \(M )\) when the particle is a distance \(r\) from the sun. (b) Let \(L\) represent the luminosity of the sun, equal to the rate at which it emits energy in electromagnetic radiation. Find the force exerted on the (totally absorbing) particle due to solar radiation pressure, remembering that the intensity of the sun's radiation also depends on the distance \(r .\) The relevant area is the cross-sectional area of the particle, \(n o t\) the total surface area of the particle. As part of your answer, explain why this is so. (c) The mass density of a typical interplanetary dust particle is about 3000 \(\mathrm{kg} / \mathrm{m}^{3}\) . Find the particle radius \(R\) such that the gravitational and radiation forces acting on the particle are equal in magnitude. The luminosity of the sun is \(3.9 \times 10^{26} \mathrm{W}\) . Does your answer depend on the distance of the particle from the sun? Why or why not? (d) Explain why dust particles with a radius less than that found in part (c) are unlikely to be found in the solar system. [Hint: Construct the ratio of the two force expressions found in parts (a) and (b).]

The electron in a hydrogen atom can be considered to be in a circular orbit with a radius of 0.0529 \(\mathrm{nm}\) and a kinetic energy of 13.6 \(\mathrm{eV}\) . If the electron behaved classically, how much energy would it radiate per second (see Challenge Problem 32.57\() ?\) What does this tell you about the use of classical physics in describing the atom?