11.6: Electromagnetic Waves (Summary)

Key Terms

direction of polarization	direction parallel to the electric field for EM waves
gamma ray (\(\displaystyle γ\) 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 \(\displaystyle γ\) -ray frequency range overlaps the upper end of the X-ray range, but \(\displaystyle γ\) rays can have the highest frequency of any electromagnetic radiation
horizontally polarized	electric field oscillations are in a horizontal plane
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 \(\displaystyle 0.74μm\) to \(\displaystyle 300μm\)
Malus’s law	\(I = I_0 \cos^2\theta\) where \(\displaystyle I_0\) is the intensity of the polarized wave before passing through the filter and \(\theta\) is the tilt angle of the filter
Maxwell’s equations	set of four equations that comprise a complete, overarching theory of electromagnetism
microwaves	electromagnetic waves with wavelengths in the range from 1 mm to 1 m; they can be produced by currents in macroscopic circuits and devices
optically active	substances that rotate the plane of polarization of light passing through them
polarized	refers to waves having the electric and magnetic field oscillations in a definite direction
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
radio waves	electromagnetic waves with wavelengths in the range from 1 mm to 100 km; 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 400 nm down to about 10 nm
unpolarized	refers to waves that are randomly polarized
vertically polarized	oscillations are in a vertical plane
visible light	narrow segment of the electromagnetic spectrum to which the normal human eye responds, from about 400 to 750 nm
x-ray	invisible, penetrating form of very high frequency electromagnetic radiation, overlapping both the ultraviolet range and the \(\displaystyle γ\)-ray range

Key Equations

Speed of EM waves	\(\displaystyle c=\frac{1}{\sqrt{ε_0μ_0}}\)
Ratio of E field to B field in electromagnetic wave	\(\displaystyle c=\frac{E}{B}\)
Energy flux (Poynting) vector	\(\displaystyle \vec{S}=\frac{1}{μ_0}\vec{E}×\vec{B}\)
Average intensity of an electromagnetic wave	\(\displaystyle I=S_{avg}=\frac{cε_0E^2_0}{2}=\frac{cB^2_0}{2μ_0}=\frac{E_0B_0}{2μ_0}\)
Malus’s law	\(\displaystyle I=I_0cos^2θ\)

Summary

Maxwell’s Equations and Electromagnetic Waves

James Clerk Maxwell (1831–1879) was one of the major contributors to physics in the nineteenth century. Although he died young, he made major contributions to the development of the kinetic theory of gases, to the understanding of color vision, and to the nature of Saturn’s rings. He is best known for having combined existing knowledge of the laws of electricity and of magnetism with insights of his own into a complete overarching electromagnetic theory, represented by Maxwell’s equations.

• 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.

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

\(\displaystyle I=\frac{cε_0E^2_0}{2}\)

where I is the average intensity in \(\displaystyle W/m^2\) and \(\displaystyle E_0\) is the maximum electric field strength of a continuous sinusoidal wave. This can also be expressed in terms of the maximum magnetic field strength \(\displaystyle B_0\) as

\(\displaystyle I=\frac{cB^2_0}{2μ_0}\)

and in terms of both electric and magnetic fields as

\(\displaystyle I=\frac{E_0B_0}{2μ_0}\).

The three expressions for \(\displaystyle I_{avg}\) are all equivalent.

The Electromagnetic Spectrum

• The relationship among the speed of propagation, wavelength, and frequency for any wave is given by \(\displaystyle v=fλ\), so that for electromagnetic waves, \(\displaystyle c=fλ\), where f is the frequency, \(\displaystyle λ\) 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.

Polarization

• Polarization is the attribute that wave oscillations have a definite direction relative to the direction of propagation of the wave. The direction of polarization is defined to be the direction parallel to the electric field of the EM wave.
• Unpolarized light is composed of many rays having random polarization directions.
• Unpolarized light can be polarized by passing it through a polarizing filter or other polarizing material. The process of polarizing light decreases its intensity by a factor of 2.
• The intensity, I, of polarized light after passing through a polarizing filter is \(\displaystyle I=I_0cos^2θ\), where \(\displaystyle I_0\) is the incident intensity and \(\displaystyle θ\) is the angle between the direction of polarization and the axis of the filter.
• Polarization is also produced by reflection.
• Brewster’s law states that reflected light is completely polarized at the angle of reflection \(\displaystyle θ_b\), known as Brewster’s angle.
• Polarization can also be produced by scattering.
• Several types of optically active substances rotate the direction of polarization of light passing through them.

Contributors and Attributions

Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).