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 (

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

microwaves
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

radar
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

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

-ray range

Key Equations

Displacement current	$I_{\mathrm{d}}=$ $\epsilon_0\frac{d\Phi_{\mathrm{E}}}{dt}}$
Gauss’s law	$\oint\vec{\mathbf{E}}\cdot d\vec{\mathbf{A}}=\frac{Q_{\mathrm{in}}}{\epsilon_0}$
Gauss’s law for magnetism	$\oint\vec{\mathbf{B}}\cdot d\vec{\mathbf{A}}=0$
Faraday’s law	$\oint\vec{\mathbf{E}}\cdot d\vec{\mathbf{s}}=-\frac{d\Phi_{\mathrm{in}}}{dt}$
Ampère-Maxwell law	$\oint\vec{\mathbf{B}}\cdot d\vec{\mathbf{s}}=\mu_0I+\epsilon_0\mu_0\frac{d\Phi_{\mathrm{E}}}{dt}$
Wave equation for plane EM wave	$\frac{\partial^2E_y}{\partial x^2}=\epsilon_0\mu_0\frac{\partial^2E_y}{\partial t^2}$
Speed of EM waves	$c=\frac{1}{\sqrt{\epsilon_0\mu_0}}$
Ratio of E field to B field in electromagnetic wave	$c=\frac{E}{B}$
Energy flux (Poynting) vector	$\vec{\mathbf{S}}=\frac{1}{\mu_0}\vec{\mathbf{E}}\times\vec{\mathbf{B}}$
Average intensity of an electromagnetic wave	$I=S_{\mathrm{avg}}=\frac{c\epsilon_0E_0^2}{2}=\frac{cB_0^2}{2\mu_0}=\frac{E_0B_0}{2\mu_0}$
Radiation pressure	$p=\left\{\begin{array}{ll}I/c&\mathrm{Perfect~absorber}\\2I/c&\mathrm{Perfect~reflector}\end{array}\right.$

Summary

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 $c=E/B,$ which implies that the magnetic field $B$
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
$I=\frac{c\epsilon_0E_0^2}{2}$
where
$I$

where is the average intensity in
$\mathrm{W/m}^2$
and

$E_0$

is the maximum electric field strength of a continuous sinusoidal wave. This can also be expressed in terms of the maximum magnetic field strength
$B_0$
as

$I=\frac{cB_0^2}{2\mu_0}$
and in terms of both electric and magnetic fields as

$I=\frac{E_0B_0}{2\mu_0}.$
The three expressions for
$I_{\mathrm{avg}}$
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 $v=f\lambda,$ so that for electromagnetic waves, $c=f\lambda,$ where $f$ is the frequency, $\lambda$ is the wavelength, and $c$ 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.

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?