## 24.1 Maxwell’s Equations: Electromagnetic Waves Predicted and Observed

- Electromagnetic waves consist of oscillating electric and magnetic fields and propagate at the speed of light $c$. They were predicted by Maxwell, who also showed that
$$c=\frac{1}{\sqrt{{\mu}_{0}{\epsilon}_{0}}},$$
where ${\mu}_{0}$ is the permeability of free space and ${\epsilon}_{0}$ is the permittivity of free space.

- 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.
- These four equations are paraphrased in this text, rather than presented numerically, and encompass the major laws of electricity and magnetism. First is Gauss’s law for electricity, second is Gauss’s law for magnetism, third is Faraday’s law of induction, including Lenz’s law, and fourth is Ampere’s law in a symmetric formulation that adds another source of magnetism—changing electric fields.

## 24.2 Production of Electromagnetic Waves

- Electromagnetic waves are created by oscillating charges (which radiate whenever accelerated) and have the same frequency as the oscillation.
- Since the electric and magnetic fields in most electromagnetic waves are perpendicular to the direction in which the wave moves, it is ordinarily a transverse wave.
- The strengths of the electric and magnetic parts of the wave are related by
$$\frac{E}{B}=\text{c},$$
which implies that the magnetic field $B$ is very weak relative to the electric field $E$.

## 24.3 The Electromagnetic Spectrum

- The relationship among the speed of propagation, wavelength, and frequency for any wave is given by ${v}_{\text{W}}=\mathrm{f\lambda}$, so that for electromagnetic waves,
$$c=\mathrm{f\lambda},$$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.
- Any electromagnetic wave produced by currents in wires is classified as a radio wave, the lowest frequency electromagnetic waves. Radio waves are divided into many types, depending on their applications, ranging up to microwaves at their highest frequencies.
- Infrared radiation lies below visible light in frequency and is produced by thermal motion and the vibration and rotation of atoms and molecules. Infrared’s lower frequencies overlap with the highest-frequency microwaves.
- Visible light is largely produced by electronic transitions in atoms and molecules, and is defined as being detectable by the human eye. Its colors vary with frequency, from red at the lowest to violet at the highest.
- Ultraviolet radiation starts with frequencies just above violet in the visible range and is produced primarily by electronic transitions in atoms and molecules.
- X-rays are created in high-voltage discharges and by electron bombardment of metal targets. Their lowest frequencies overlap the ultraviolet range but extend to much higher values, overlapping at the high end with gamma rays.
- Gamma rays are nuclear in origin and are defined to include the highest-frequency electromagnetic radiation of any type.

## 24.4 Energy in Electromagnetic Waves

- The energy carried by any wave is proportional to its amplitude squared. For electromagnetic waves, this means intensity can be expressed as
$${I}_{\text{ave}}=\frac{{\mathrm{c\epsilon}}_{0}{E}_{0}^{2}}{2},$$
where ${I}_{\text{ave}}$ is the average intensity in ${\text{W/m}}^{2}$, 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
$${I}_{\text{ave}}=\frac{{\text{cB}}_{0}^{2}}{{\mathrm{2\mu}}_{0}}$$
and in terms of both electric and magnetic fields as

$${I}_{\text{ave}}=\frac{{E}_{0}{B}_{0}}{{\mathrm{2\mu}}_{0}}.$$ - The three expressions for ${I}_{\text{ave}}$ are all equivalent.