# 16.S: Electromagnetic Waves (Summary)

- Page ID
- 10313

# 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 (\(\displaystyle γ\) 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 |

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 \(\displaystyle 300μm\) |

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 |

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 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 |

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

Displacement current | \(\displaystyle I_d=ε_0\frac{dΦ_E}{dt}\) |

Gauss’s law | \(\displaystyle ∮\vec{E}⋅d\vec{A}=\frac{Q_{in}}{ε_0}\) |

Gauss’s law for magnetism | \(\displaystyle ∮\vec{B}⋅d\vec{A}=0\) |

Faraday’s law | \(\displaystyle ∮\vec{E}⋅d\vec{s}=−\frac{dΦ_m}{dt}\) |

Ampère-Maxwell law | \(\displaystyle ∮\vec{B}⋅d\vec{s}=μ_0I+ε_0μ_0\frac{dΦ_E}{dt}\) |

Wave equation for plane EM wave | \(\displaystyle \frac{∂^2E_y}{∂x^2}=ε_0μ_0\frac{∂^2E_y}{∂t^2}\) |

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 | \(\displaystyle I=S_{avg}=\frac{cε_0E^2_0}{2}=\frac{cB^2_0}{2μ_0}=\frac{E_0B_0}{2μ_0}\) |

Radiation pressure | \(\displaystyle p= \begin{cases} I/c & Perfect absorber \\ 2I/c & Perfect reflector \end{cases}\) |

# Summary

## 16.1: 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.

## 16.2: Plane Electromagnetic Waves

Mechanical waves travel through a medium such as a string, water, or air. Perhaps the most significant prediction of Maxwell’s equations is the existence of combined electric and magnetic (or electromagnetic) fields that propagate through space as electromagnetic waves. Because Maxwell’s equations hold in free space, the predicted electromagnetic waves, unlike mechanical waves, do not require a medium for their propagation.

- 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 \(\displaystyle c=E/B\), which implies that the magnetic field
*B*is very weak relative to the electric field*E*. - Accelerating charges create electromagnetic waves (for example, an oscillating current in a wire produces electromagnetic waves with the same frequency as the oscillation).

## 16.3: 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

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

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.

## 16.4: 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.

## 16.5: 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.

# Contributors

Paul Peter Urone (Professor Emeritus at California State University, Sacramento) and Roger Hinrichs (State University of New York, College at Oswego) with Contributing Authors: Kim Dirks (University of Auckland) and Manjula Sharma (University of Sydney). This work is licensed by OpenStax University Physics under a Creative Commons Attribution License (by 4.0).