Chapter 31

Which of the following statements concerning electromagnetic waves are incorrect? (Select all that apply.) a) Electromagnetic waves in vacuum travel at the speed of light. b) The magnitudes of the electric field and the magnetic field are equal. c) Only the electric field vector is perpendicular to the direction of the wave's propagation. d) Both the electric field vector and the magnetic field vector are perpendicular to the direction of propagation. e) An electromagnetic wave carries energy only when \(E=B\).

Which of the following exerts the largest amount of radiation pressure? a) a \(1-\mathrm{mW}\) laser pointer on a \(2-\mathrm{mm}\) -diameter spot \(1 \mathrm{~m}\) away b) a 200-W light bulb on a 4 -mm-diameter spot \(10 \mathrm{~m}\) away c) a 100 -W light bulb on a 2 -mm-diameter spot 4 m away d) a 200 - \(\mathrm{W}\) light bulb on a 2 -mm-diameter spot \(5 \mathrm{~m}\) away e) All of the above exert the same pressure.

It is speculated that isolated magnetic "charges" (magnetic monopoles) may exist somewhere in the universe. Which of Maxwell's equations, (1) Gauss's Law for Electric Fields, (2) Gauss's Law for Magnetic Fields, (3) Faraday's Law of Induction, and/or (4) the MaxwellAmpere Law, would be altered by the existence of magnetic monopoles? a) only (2) c) (2) and (3) b) (1) and (2) d) only (3)

31.8 According to Gauss's Law for Magnetic Fields, all magnetic field lines form a complete loop. Therefore, the direction of the magnetic field \(\vec{B}\) points from _________ pole to ________ pole outside of an ordinary bar magnet and from ____ pole to pole _______ inside the magnet. a) north, south, north, south b) north, south, south, north c) south, north, south, north d) south, north, north, south

A dipole antenna is located at the origin with its axis along the \(z\) -axis. As electric current oscillates up and down the antenna, polarized electromagnetic radiation travels away from the antenna along the positive \(y\) -axis. What are the possible directions of electric and magnetic fields at point \(A\) on the \(y\) -axis? Explain.

If two communication signals were sent at the same time to the Moon, one via radio waves and one via visible light, which one would arrive at the Moon first?

Show that Ampere's Law is not necessarily consistent if the surface through which the flux is to be calculated is a closed surface, but that the Maxwell- Ampere Law always is. (Hence, Maxwell's introduction of his law of induction and the displacement current are not optional; they are logically necessary.) Show also that Faraday's Law of Induction does not suffer from this consistency problem.

Practically everyone who has studied the electromagnetic spectrum has wondered how the world would appear if we could see over a range of frequencies of the ten octaves over which we can hear rather than the less than one octave over which we can see. (An octave refers to a factor of 2 in frequency.) But this is fundamentally impossible. Why?

Electromagnetic waves from a small, isotropic source are not plane waves, which have constant maximum amplitudes. a) How does the maximum amplitude of the electric field of radiation from a small, isotropic source vary with distance from the source? b) Compare this with the electrostatic field of a point charge.

Two polarizing filters are crossed at \(90^{\circ}\), so when light is shined from behind the pair of filters, no light passes through. A third filter is inserted between the two, initially aligned with one of them. Describe what happens as the intermediate filter is rotated through an angle of \(360^{\circ} .\)

An electric field of magnitude \(200.0 \mathrm{~V} / \mathrm{m}\) is directed perpendicular to a circular planar surface with radius \(6.00 \mathrm{~cm}\). If the electric field increases at a rate of \(10.0 \mathrm{~V} /(\mathrm{m} \mathrm{s}),\) determine the magnitude and the direction of the magnetic field at a radial distance \(10.0 \mathrm{~cm}\) away from the center of the circular area.

A wire of radius \(1.0 \mathrm{~mm}\) carries a current of 20.0 A. The wire is connected to a parallel plate capacitor with circular plates of radius \(R=4.0 \mathrm{~cm}\) and a separation between the plates of \(s=2.0 \mathrm{~mm} .\) What is the magnitude of the magnetic field due to the changing electric field at a point that is a radial distance of \(r=1.0 \mathrm{~cm}\) from the center of the parallel plates? Neglect edge effects.

The current flowing in a solenoid that is \(20.0 \mathrm{~cm}\) long and has a radius of \(2.00 \mathrm{~cm}\) and 500 turns decreases from \(3.00 \mathrm{~A}\) to \(1.00 \mathrm{~A}\) in \(0.1 .00 \mathrm{~s} .\) Determine the magnitude of the induced electric field inside the solenoid \(1.00 \mathrm{~cm}\) from its center.

The voltage across a cylindrical conductor of radius \(r\), length \(L\), and resistance \(R\) varies with time. The timevarying voltage causes a time-varying current, \(i\), to flow in the cylinder. Show that the displacement current equals \(\epsilon_{0} \rho d i / d t,\) where \(\rho\) is the resistivity of the conductor.

Determine the distance in feet that light can travel in vacuum during \(1.00 \mathrm{~ns}\).

How long does it take light to travel from the Moon to the Earth? From the Sun to the Earth? From Jupiter to the Earth?

Alice made a telephone call from her home telephone in New York to her fiancé stationed in Baghdad, about \(10,000 \mathrm{~km}\) away, and the signal was carried on a telephone cable. The following day, Alice called her fiancé again from work using her cell phone, and the signal was transmitted via a satellite \(36,000 \mathrm{~km}\) above the Earth's surface, halfway between New York and Baghdad. Estimate the time taken for the signals sent by (a) the telephone cable and (b) via the satellite to reach Baghdad, assuming that the signal speed in both cases is the same as speed of light, \(c .\) Would there be a noticeable delay in either case?

Electric and magnetic fields in many materials can be analyzed using the same relationships as for fields in vacuum, only substituting relative values of the permittivity and the permeability, \(\epsilon=\kappa \epsilon_{0}\) and \(\mu=\kappa_{\mathrm{m}} \mu_{0},\) for their vacuum values, where \(\kappa\) is the dielectric constant and \(\kappa_{\mathrm{m}}\) the relative permeability of the material. Calculate the ratio of the speed of electromagnetic waves in vacuum to their speed in such a material.

The wavelength range for visible light is \(400 \mathrm{nm}\) to \(700 \mathrm{nm}\) (see Figure 31.10 ) in air. What is the frequency range of visible light?

The antenna of a cell phone is a straight rod \(8.0 \mathrm{~cm}\) long. Calculate the operating frequency of the signal from this phone, assuming that the antenna length is \(\frac{1}{4}\) of the wavelength of the signal.

Suppose an RLC circuit in resonance is used to produce a radio wave of wavelength \(150 \mathrm{~m}\). If the circuit has a 2.0 -pF capacitor, what size inductor is used?

Three FM radio stations covering the same geographical area broadcast at frequencies \(91.1,91.3,\) and \(91.5 \mathrm{MHz},\) respectively. What is the maximum allowable wavelength width of the band-pass filter in a radio receiver so that the FM station 91.3 can be played free of interference from FM 91.1 or FM 91.5? Use \(c=3.00 \cdot 10^{8} \mathrm{~m} / \mathrm{s}\), and calculate the wavelength to an uncertainty of \(1 \mathrm{~mm} .\)

A monochromatic point source of light emits \(1.5 \mathrm{~W}\) of electromagnetic power uniformly in all directions. Find the Poynting vector at a point situated at each of the following locations: a) \(0.30 \mathrm{~m}\) from the source b) \(0.32 \mathrm{~m}\) from the source c) \(1.00 \mathrm{~m}\) from the source

Consider an electron in a hydrogen atom, which is \(0.050 \mathrm{nm}\) from the proton in the nucleus. a) What electric field does the electron experience? b) In order to produce an electric field whose root-meansquare magnitude is the same as that of the field in part (a), what intensity must a laser light have?

Calculate the average value of the Poynting vector, \(S_{\text {ave }}\) for an electromagnetic wave having an electric field of amplitude \(100 . \mathrm{V} / \mathrm{m}\) a) What is the average energy density of this wave? b) How large is the amplitude of the magnetic field?

The most intense beam of light that can propagate through dry air must have an electric field whose maximum amplitude is no greater than the breakdown value for air: \(E_{\max }^{\operatorname{air}}=3.0 \cdot 10^{6} \mathrm{~V} / \mathrm{m},\) assuming that this value is unaffected by the frequency of the wave. a) Calculate the maximum amplitude the magnetic field of this wave can have. b) Calculate the intensity of this wave. c) What happens to a wave more intense than this?

A continuous-wave (cw) argon-ion laser beam has an average power of \(10.0 \mathrm{~W}\) and a beam diameter of \(1.00 \mathrm{~mm}\). Assume that the intensity of the beam is the same throughout the cross section of the beam (which is not true, as the actual distribution of intensity is a Gaussian function). a) Calculate the intensity of the laser beam. Compare this with the average intensity of sunlight at Earth's surface \(\left(1400 . \mathrm{W} / \mathrm{m}^{2}\right)\) b) Find the root-mean-square electric field in the laser beam. c) Find the average value of the Poynting vector over time. d) If the wavelength of the laser beam is \(514.5 \mathrm{nm}\) in vacuum, write an expression for the instantaneous Poynting vector, where the instantaneous Poynting vector is zero at \(t=0\) and \(x=0\) e) Calculate the root-mean-square value of the magnetic field in the laser beam.

A voltage, \(V\), is applied across a cylindrical conductor of radius \(r\), length \(L\), and resistance \(R\). As a result, a current, \(i\), is flowing through the conductor, which gives rise to a magnetic field, \(B\). The conductor is placed along the \(y\) -axis, and the current is flowing in the positive \(y\) -direction. Assume that the electric field is uniform throughout the conductor. a) Find the magnitude and the direction of the Poynting vector at the surface of the conductor. b) Show that \(\int \vec{S} \cdot d \vec{A}=i^{2} R\)

Scientists have proposed using the radiation pressure of sunlight for travel to other planets in the Solar System. If the intensity of the electromagnetic radiation produced by the Sun is about \(1.40 \mathrm{~kW} / \mathrm{m}^{2}\) near the Earth, what size would a sail have to be to accelerate a spaceship with a mass of 10.0 metric tons at \(1.00 \mathrm{~m} / \mathrm{s}^{2} ?\) a) Assume that the sail absorbs all the incident radiation. b) Assume that the sail perfectly reflects all the incident radiation.

A solar sail is a giant circle (with a radius \(R=10.0 \mathrm{~km}\) ) made of a material that is perfectly reflecting on one side and totally absorbing on the other side. In deep space, away from other sources of light, the cosmic microwave background will provide the primary source of radiation incident on the sail. Assuming that this radiation is that of an ideal black body at \(T=2.725 \mathrm{~K},\) calculate the net force on the sail due to its reflection and absorption.

A tiny particle of density \(2000 . \mathrm{kg} / \mathrm{m}^{3}\) is at the same distance from the Sun as the Earth is \(\left(1.50 \cdot 10^{11} \mathrm{~m}\right)\). Assume that the particle is spherical and perfectly reflecting. What would its radius have to be for the outward radiation pressure on it to be \(1.00 \%\) of the inward gravitational attraction of the Sun? (Take the Sun's mass to be \(\left.2.00 \cdot 10^{30} \mathrm{~kg} .\right)\)

Silica aerogel, an extremely porous, thermally insulating material made of silica, has a density of \(1.00 \mathrm{mg} / \mathrm{cm}^{3}\). A thin circular slice of aerogel has a diameter of \(2.00 \mathrm{~mm}\) and a thickness of \(0.10 \mathrm{~mm}\). a) What is the weight of the aerogel slice (in newtons)? b) What is the intensity and radiation pressure of a \(5.00-\mathrm{mW}\) laser beam of diameter \(2.00 \mathrm{~mm}\) on the sample? c) How many \(5.00-\mathrm{mW}\) lasers with a beam diameter of \(2.00 \mathrm{~mm}\) would be needed to make the slice float in the Earth's gravitational field? Use \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\)

Two polarizers are out of alignment by \(30.0^{\circ} .\) If light of intensity \(1.00 \mathrm{~W} / \mathrm{m}^{2}\) and initially polarized halfway between the polarizing angles of the two filters passes through the two filters, what is the intensity of the transmitted light?

A 10.0 -mW vertically polarized laser beam passes through a polarizer whose polarizing angle is \(30.0^{\circ}\) from the horizontal. What is the power of the laser beam when it emerges from the polarizer?

Unpolarized light of intensity \(I_{0}\) is incident on a series of five polarizers, each rotated \(10.0^{\circ}\) from the preceding one. What fraction of the incident light will pass through the series?

A laser beam takes 50.0 ms to be reflected back from a totally reflecting sail on a spacecraft. How far away is the sail?

A house with a south-facing roof has photovoltaic panels on the roof. The photovoltaic panels have an efficiency of \(10.0 \%\) and occupy an area with dimensions \(3.00 \mathrm{~m}\) by \(8.00 \mathrm{~m} .\) The average solar radiation incident on the panels is \(300 . \mathrm{W} / \mathrm{m}^{2}\), averaged over all conditions for a year. How many kilowatt hours of electricity will the solar panels generate in a 30 -day month?

What is the radiation pressure due to Betelgeuse (which has a luminosity, or power output, 10,000 times that of the Sun) at a distance equal to that of Uranus's orbit from it?

What is the wavelength of the electromagnetic waves used for cell phone communications in the 850 -MHz band?

As shown in the figure, sunlight is coming straight down (negative \(z\) -direction) on a solar panel (of length \(L=1.40 \mathrm{~m}\) and width \(W=0.900 \mathrm{~m}\) ) on the Mars rover Spir- it. The amplitude of the electric field in the solar radiation is \(673 \mathrm{~V} / \mathrm{m}\) and is uniform (the radiation has the same amplitude everywhere). If the solar panel has an efficiency of \(18.0 \%\) in converting solar radiation into electrical power, how much average power can the panel generate?

A \(14.9-\mu F\) capacitor, a \(24.3-\mathrm{k} \Omega\) resistor, a switch, and a 25.-V battery are connected in series. What is the rate of change of the electric field between the plates of the capacitor at \(t=0.3621 \mathrm{~s}\) after the switch is closed? The area of the plates is \(1.00 \mathrm{~cm}^{2}\) .

What is the distance between successive heating antinodes in a microwave oven's cavity? A microwave oven typically operates at a frequency of \(2.4 \mathrm{GHz}\).

A \(5.00-\mathrm{mW}\) laser pointer has a beam diameter of \(2.00 \mathrm{~mm}\) a) What is the root-mean-square value of the electric field in this laser beam? b) Calculate the total electromagnetic energy in \(1.00 \mathrm{~m}\) of this laser beam.

At the surface of the Earth, the Sun delivers an estimated \(1.00 \mathrm{~kW} / \mathrm{m}^{2}\) of energy. Suppose sunlight hits a \(10.0 \mathrm{~m}\) by \(30.0 \mathrm{~m}\) roof at an angle of \(90.0^{\circ}\) a) Estimate the total power incident on the roof. b) Find the radiation pressure on the roof.

A resistor consists of a solid cylinder of radius \(r\) and length \(L\). The resistor has resistance \(R\) and is carrying current \(i\). Use the Poynting vector to calculate the power radiated out of the surface of the resistor.

Quantum theory says that electromagnetic waves actually consist of discrete packets-photons-each with energy \(E=\hbar \omega,\) where \(\hbar=1.054573 \cdot 10^{-34} \mathrm{~J} \mathrm{~s}\) is Planck's reduced constant and \(\omega\) is the angular frequency of the wave. a) Find the momentum of a photon. b) Find the angular momentum of a photon. Photons are circularly polarized; that is, they are described by a superposition of two plane-polarized waves with equal field amplitudes, equal frequencies, and perpendicular polarizations, one-quarter of a cycle \(\left(90^{\circ}\right.\) or \(\pi / 2\) rad \()\) out of phase, so the electric and magnetic field vectors at any fixed point rotate in a circle with the angular frequency of the waves. It can be shown that a circularly polarized wave of energy \(U\) and angular frequency \(\omega\) has an angular momentum of magnitude \(L=U / \omega .\) (The direction of the angular momentum is given by the thumb of the right hand, when the fingers are curled in the direction in which the field vectors circulate. c) The ratio of the angular momentum of a particle to \(\hbar\) is its spin quantum number. Determine the spin quantum number of the photon.

.An industrial carbon dioxide laser produces a beam of radiation with average power of \(6.00 \mathrm{~kW}\) at a wavelength of \(10.6 \mu \mathrm{m}\). Such a laser can be used to cut steel up to \(25 \mathrm{~mm}\) thick. The laser light is polarized in the \(x\) -direction, travels in the positive \(z\) -direction, and is collimated (neither diverging or converging) at a constant diameter of \(100.0 \mu \mathrm{m} .\) Write the equations for the laser light's electric and magnetic fields as a function of time and of position \(z\) along the beam. Recall that \(\vec{E}\) and \(\vec{B}\) are vectors. Leave the overall phase unspecified, but be sure to check the relative phase between \(\vec{E}\) and \(\vec{B}\) .