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Chapter 13 Review

Key Terms

displacement current
extra term in Maxwell’s equations that is analogous to a real current but accounts for a changing electric field producing a magnetic field, even when the real current is present

gamma ray (

extremely high frequency electromagnetic radiation emitted by the nucleus of an atom, either from natural nuclear decay or induced nuclear processes in nuclear reactors and weapons; the lower end of the

ray frequency range overlaps the upper end of the X-ray range, but

rays can have the highest frequency of any electromagnetic radiation

infrared radiation
region of the electromagnetic spectrum with a frequency range that extends from just below the red region of the visible light spectrum up to the microwave region, or from


Maxwell’s equations
set of four equations that comprise a complete, overarching theory of electromagnetism

electromagnetic waves with wavelengths in the range from


; they can be produced by currents in macroscopic circuits and devices

Poynting vector
vector equal to the cross product of the electric-and magnetic fields, that describes the flow of electromagnetic energy through a surface

common application of microwaves; radar can determine the distance to objects as diverse as clouds and aircraft, as well as determine the speed of a car or the intensity of a rainstorm

radiation pressure
force divided by area applied by an electromagnetic wave on a surface

radio waves
electromagnetic waves with wavelengths in the range from


; they are produced by currents in wires and circuits and by astronomical phenomena

thermal agitation
thermal motion of atoms and molecules in any object at a temperature above absolute zero, which causes them to emit and absorb radiation

ultraviolet radiation
electromagnetic radiation in the range extending upward in frequency from violet light and overlapping with the lowest X-ray frequencies, with wavelengths from

down to about

visible light
narrow segment of the electromagnetic spectrum to which the normal human eye responds, from about


invisible, penetrating form of very high frequency electromagnetic radiation, overlapping both the ultraviolet range and the

-ray range

Key Equations

Displacement current
Gauss’s law
Gauss’s law for magnetism
Faraday’s law
Ampère-Maxwell law
Wave equation for plane EM wave
Speed of EM waves
Ratio of E field to B field in electromagnetic wave
Energy flux (Poynting) vector
Average intensity of an electromagnetic wave
Radiation pressure


13.1 Maxwell’s Equations and Electromagnetic Waves

  • Maxwell’s prediction of electromagnetic waves resulted from his formulation of a complete and symmetric theory of electricity and magnetism, known as Maxwell’s equations.
  • The four Maxwell’s equations together with the Lorentz force law encompass the major laws of electricity and magnetism. The first of these is Gauss’s law for electricity; the second is Gauss’s law for magnetism; the third is Faraday’s law of induction (including Lenz’s law); and the fourth is Ampère’s law in a symmetric formulation that adds another source of magnetism, namely changing electric fields.
  • The symmetry introduced between electric and magnetic fields through Maxwell’s displacement current explains the mechanism of electromagnetic wave propagation, in which changing magnetic fields produce changing electric fields and vice versa.
  • Although light was already known to be a wave, the nature of the wave was not understood before Maxwell. Maxwell’s equations also predicted electromagnetic waves with wavelengths and frequencies outside the range of light. These theoretical predictions were first confirmed experimentally by Heinrich Hertz.

13.2 Plane Electromagnetic Waves

  • Maxwell’s equations predict that the directions of the electric and magnetic fields of the wave, and the wave’s direction of propagation, are all mutually perpendicular. The electromagnetic wave is a transverse wave.
  • The strengths of the electric and magnetic parts of the wave are related by which implies that the magnetic field
  • Accelerating charges create electromagnetic waves (for example, an oscillating current in a wire produces electromagnetic waves with the same frequency as the oscillation).

13.3 Energy Carried by Electromagnetic Waves

  • The energy carried by any wave is proportional to its amplitude squared. For electromagnetic waves, this means intensity can be expressed as


    where is the average intensity in


    is the maximum electric field strength of a continuous sinusoidal wave. This can also be expressed in terms of the maximum magnetic field strength


    and in terms of both electric and magnetic fields as

    The three expressions for

    are all equivalent.

13.4 Momentum and Radiation Pressure

  • Electromagnetic waves carry momentum and exert radiation pressure.
  • The radiation pressure of an electromagnetic wave is directly proportional to its energy density.
  • The pressure is equal to twice the electromagnetic energy intensity if the wave is reflected and equal to the incident energy intensity if the wave is absorbed.

13.5 The Electromagnetic Spectrum

  • The relationship among the speed of propagation, wavelength, and frequency for any wave is given by so that for electromagnetic waves, where is the frequency, is the wavelength, and is the speed of light.
  • The electromagnetic spectrum is separated into many categories and subcategories, based on the frequency and wavelength, source, and uses of the electromagnetic waves.

Answers to Check Your Understanding

13.1 It is greatest immediately after the current is switched on. The displacement current and the magnetic field from it are proportional to the rate of change of electric field between the plates, which is greatest when the plates first begin to charge.

13.2 No. The changing electric field according to the modified version of Ampère’s law would necessarily induce a changing magnetic field.

13.3 (1) Faraday’s law, (2) the Ampère-Maxwell law

13.4 a. The directions of wave propagation, of the

field, and of

field are all mutually perpendicular. b. The speed of the electromagnetic wave is the speed of light

independent of frequency. c. The ratio of electric and magnetic field amplitudes is

13.5 Its acceleration would decrease because the radiation force is proportional to the intensity of light from the Sun, which decreases with distance. Its speed, however, would not change except for the effects of gravity from the Sun and planets.

13.6 They fall into different ranges of wavelength, and therefore also different corresponding ranges of frequency.

Conceptual Questions

13.1 Maxwell’s Equations and Electromagnetic Waves

1. Explain how the displacement current maintains the continuity of current in a circuit containing a capacitor.

2. Describe the field lines of the induced magnetic field along the edge of the imaginary horizontal cylinder shown below if the cylinder is in a spatially uniform electric field that is horizontal, pointing to the right, and increasing in magnitude.

field lines of the induced magnetic field

3. Why is it much easier to demonstrate in a student lab that a changing magnetic field induces an electric field than it is to demonstrate that a changing electric field produces a magnetic field?

13.2 Plane Electromagnetic Waves

4. If the electric field of an electromagnetic wave is oscillating along the

-axis and the magnetic field is oscillating along the

-axis, in what possible direction is the wave traveling?

5. In which situation shown below will the electromagnetic wave be more successful in inducing a current in the wire? Explain.

electromagnetic wave

6. In which situation shown below will the electromagnetic wave be more successful in inducing a current in the loop? Explain.

electromagnetic wave

7. Under what conditions might wires in a circuit where the current flows in only one direction emit electromagnetic waves?

8. Shown below is the interference pattern of two radio antennas broadcasting the same signal. Explain how this is analogous to the interference pattern for sound produced by two speakers. Could this be used to make a directional antenna system that broadcasts preferentially in certain directions? Explain.

interference pattern of two radio antennas

13.3 Energy Carried by Electromagnetic Waves

9. When you stand outdoors in the sunlight, why can you feel the energy that the sunlight carries, but not the momentum it carries?

10. How does the intensity of an electromagnetic wave depend on its electric field? How does it depend on its magnetic field?

11. What is the physical significance of the Poynting vector?

12. A

helium-neon laser transmits a continuous beam of red light of cross-sectional area

If the beam does not diverge appreciably, how would its rms electric field vary with distance from the laser? Explain.

13.4 Momentum and Radiation Pressure

13. Why is the radiation pressure of an electromagnetic wave on a perfectly reflecting surface twice as large as the pressure on a perfectly absorbing surface?

14. Why did the early Hubble Telescope photos of Comet Ison approaching Earth show it to have merely a fuzzy coma around it, and not the pronounced double tail that developed later (see below)?

(Figure 13.6.1)

Hubble Telescope photos of Comet Ison
Figure 13.6.1 (credit: ESA, Hubble)

15. (a) If the electric field and magnetic field in a sinusoidal plane wave were interchanged, in which direction relative to before would the energy propagate?

(b) What if the electric and the magnetic fields were both changed to their negatives?

13.5 The Electromagnetic Spectrum

16. Compare the speed, wavelength, and frequency of radio waves and X-rays traveling in a vacuum.

17. Accelerating electric charge emits electromagnetic radiation. How does this apply in each case: (a) radio waves, (b) infrared radiation.

18. Compare and contrast the meaning of the prefix “micro” in the names of SI units in the term microwaves.

19. Part of the light passing through the air is scattered in all directions by the molecules comprising the atmosphere. The wavelengths of visible light are larger than molecular sizes, and the scattering is strongest for wavelengths of light closest to sizes of molecules.

(a) Which of the main colors of light is scattered the most? (b) Explain why this would give the sky its familiar background color at midday.

20. When a bowl of soup is removed from a microwave oven, the soup is found to be steaming hot, whereas the bowl is only warm to the touch. Discuss the temperature changes that have occurred in terms of energy transfer.

21. Certain orientations of a broadcast television antenna give better reception than others for a particular station. Explain.

22. What property of light corresponds to loudness in sound?

23. Is the visible region a major portion of the electromagnetic spectrum?

24. Can the human body detect electromagnetic radiation that is outside the visible region of the spectrum?

25. Radio waves normally have their


fields in specific directions, whereas visible light usually has its


fields in random and rapidly changing directions that are perpendicular to each other and to the propagation direction. Can you explain why?

26. Give an example of resonance in the reception of electromagnetic waves.

27. Illustrate that the size of details of an object that can be detected with electromagnetic waves is related to their wavelength, by comparing details observable with two different types (for example, radar and visible light).

28. In which part of the electromagnetic spectrum are each of these waves:





29. In what range of electromagnetic radiation are the electromagnetic waves emitted by power lines in a country that uses

ac current?

30. If a microwave oven could be modified to merely tune the waves generated to be in the infrared range instead of using microwaves, how would this affect the uneven heating of the oven?

31. A leaky microwave oven in a home can sometimes cause interference with the homeowner’s WiFi system. Why?

32. When a television news anchor in a studio speaks to a reporter in a distant country, there is sometimes a noticeable lag between when the anchor speaks in the studio and when the remote reporter hears it and replies. Explain what causes this delay.


13.1 Maxwell’s Equations and Electromagnetic Waves

33. Show that the magnetic field at a distance r from the axis of two circular parallel plates, produced by placing charge

on the plates is

34. Express the displacement current in a capacitor in terms of the capacitance and the rate of change of the voltage across the capacitor.

35. A potential difference

is maintained across a parallel-plate capacitor with capacitance

consisting of two circular parallel plates. A thin wire with resistance

connects the centres of the two plates, allowing charge to leak between plates while they are charging.

(a) Obtain expressions for the leakage current

in the thin wire. Use these results to obtain an expression for the current

in the wires connected to the capacitor.

(b) Find the displacement current in the space between the plates from the changing electric field between the plates.

(c) Compare

with the sum of the displacement current

and resistor current

between the plates, and explain why the relationship you observe would be expected.

36. Suppose the parallel-plate capacitor shown below is accumulating charge at a rate of

What is the induced magnetic field at a distance of

from the capacitor?

parallel-plate capacitor

37. The potential difference

between parallel plates shown above is instantaneously increasing at a rate of

What is the displacement current between the plates if the separation of the plates is

and they have an area of


38. A parallel-plate capacitor has a plate area of

and a separation of

What must be must be the angular frequency

for a voltage


to produce a maximum displacement induced current of

between the plates?

39. The voltage across a parallel-plate capacitor with area

and separation

varies sinusoidally as


is in seconds. Find the displacement current between the plates.

40. The voltage across a parallel-plate capacitor with area

and separation

varies with time



is a constant. Find the displacement current between the plates.

16.2 Plane Electromagnetic Waves

41. If the Sun suddenly turned off, we would not know it until its light stopped coming. How long would that be, given that the Sun is


42. What is the maximum electric field strength in an electromagnetic wave that has a maximum magnetic field strength of


times Earth’s magnetic field)?

43. An electromagnetic wave has a frequency of

What is its wavelength in vacuum?

44. If electric and magnetic field strengths vary sinusoidally in time at frequency

being zero at



(a) When are the field strengths next equal to zero? (b) When do they reach their most negative value? (c) How much time is needed for them to complete one cycle?

45. The electric field of an electromagnetic wave traveling in vacuum is described by the following wave function:


is the wavenumber in

is in

is in

Find the following quantities:

(a) amplitude

(b) frequency

(c) wavelength

(d) the direction of the travel of the wave

(e) the associated magnetic field wave

46. A plane electromagnetic wave of frequency

moves in the positive

-axis direction such that its electric field is pointed along the

-axis. The amplitude of the electric field is

The start of time is chosen so that at

the electric field has a value

at the origin. (a) Write the wave function that will describe the electric field wave. (b) Find the wave function that will describe the associated magnetic field wave.

47. The following represents an electromagnetic wave traveling in the direction of the positive


The wave is passing through a wide tube of circular cross-section of radius

whose axis is along the

-axis. Find the expression for the displacement current through the tube.

13.3 Energy Carried by Electromagnetic Waves

48. While outdoors on a sunny day, a student holds a large convex lens of radius

above a sheet of paper to produce a bright spot on the paper that is

in radius, rather than a sharp focus. By what factor is the electric field in the bright spot of light related to the electric field of sunlight leaving the side of the lens facing the paper?

49. A plane electromagnetic wave travels northward. At one instant, its electric field has a magnitude of

and points eastward. What are the magnitude and direction of the magnetic field at this instant?

50. The electric field of an electromagnetic wave is given by

Write the equations for the associated magnetic field and Poynting vector.

51. A radio station broadcasts at a frequency of

At a receiver some distance from the antenna, the maximum magnetic field of the electromagnetic wave detected is

(a) What is the maximum electric field? (b) What is the wavelength of the electromagnetic wave?

52. The filament in a clear incandescent light bulb radiates visible light at a power of

Model the glass part of the bulb as a sphere of radius

and calculate the amount of electromagnetic energy from visible light inside the bulb.

53. At what distance does a

lightbulb produce the same intensity of light as a

lightbulb produces

away? (Assume both have the same efficiency for converting electrical energy in the circuit into emitted electromagnetic energy.)

54. An incandescent light bulb emits only

of its power as visible light. What is the rms electric field of the emitted light at a distance of

from the bulb?

55. A

lightbulb emits

of its energy as electromagnetic radiation. What is the magnitude of the average Poynting vector

from the bulb?

56. A small helium-neon laser has a power output of

What is the electromagnetic energy in a

length of the beam?

57. At the top of Earth’s atmosphere, the time-averaged Poynting vector associated with sunlight has a magnitude of about

(a) What are the maximum values of the electric and magnetic fields for a wave of this intensity? (b) What is the total power radiated by the sun? Assume that the Earth is

from the Sun and that sunlight is composed of electromagnetic plane waves.

58. The magnetic field of a plane electromagnetic wave moving along the

-axis is given by



(a) Write an expression for the electric field associated with the wave. (b) What are the frequency and the wavelength of the wave? (c) What is its average Poynting vector?

59. What is the intensity of an electromagnetic wave with a peak electric field strength of


60. Assume the helium-neon lasers commonly used in student physics laboratories have power outputs of

(a) If such a laser beam is projected onto a circular spot

in diameter, what is its intensity? (b) Find the peak magnetic field strength. (c) Find the peak electric field strength.

61. An AM radio transmitter broadcasts

of power uniformly in all directions. (a) Assuming all of the radio waves that strike the ground are completely absorbed, and that there is no absorption by the atmosphere or other objects, what is the intensity

away? (Hint: Half the power will be spread over the area of a hemisphere.) (b) What is the maximum electric field strength at this distance?

62. Suppose the maximum safe intensity of microwaves for human exposure is taken to be

(a) If a radar unit leaks

of microwaves (other than those sent by its antenna) uniformly in all directions, how far away must you be to be exposed to an intensity considered to be safe? Assume that the power spreads uniformly over the area of a sphere with no complications from absorption or reflection. (b) What is the maximum electric field strength at the safe intensity? (Note that early radar units leaked more than modern ones do. This caused identifiable health problems, such as cataracts, for people who worked near them.)

63. A

-diameter university communications satellite dish receives TV signals that have a maximum electric field strength (for one channel) of

(see below). (a) What is the intensity of this wave? (b) What is the power received by the antenna? (c) If the orbiting satellite broadcasts uniformly over an area of

(a large fraction of North America), how much power does it radiate?

university communications satellite dish

64. Lasers can be constructed that produce an extremely high intensity electromagnetic wave for a brief time—called pulsed lasers. They are used to initiate nuclear fusion, for example. Such a laser may produce an electromagnetic wave with a maximum electric field strength of