6 Electromagnetic Waves

Builds from Maxwell’s equations to electromagnetic-wave behavior, then explains how these waves transport energy and how their intensity can be calculated.

Maxwell’s equations and changing fields

Maxwell’s equations connect electric and magnetic fields with charge and current. In SI units, their integral forms are:

\oint_S \mathbf{E}\cdot d\mathbf{A}=\frac{Q_{\text{enc}}}{\varepsilon_0} \qquad\text{(Gauss’s law)}

\oint_S \mathbf{B}\cdot d\mathbf{A}=0 \qquad\text{(Gauss’s law for magnetism)}

\oint_C \mathbf{E}\cdot d\boldsymbol{\ell}=-\frac{d\Phi_B}{dt} \qquad\text{(Faraday’s law)}

\oint_C \mathbf{B}\cdot d\boldsymbol{\ell}=\mu_0 I_{\text{enc}}+\mu_0\varepsilon_0\frac{d\Phi_E}{dt} \qquad\text{(Ampère–Maxwell law)}

The first two equations apply to closed surfaces; the last two apply around closed loops. Here, $\mathbf{E}$ is the electric field, $\mathbf{B}$ is the magnetic field, $Q_{\text{enc}}$ is enclosed charge, and $I_{\text{enc}}$ is enclosed current. The quantities $\Phi_E$ and $\Phi_B$ are electric and magnetic flux; $\varepsilon_0$ and $\mu_0$ are the vacuum permittivity and permeability.

Two relationships are especially important for understanding waves. says that a changing magnetic field produces an electric field. The says that a changing electric field can produce a magnetic field, even where no conduction current flows. Together, these effects allow changing fields to sustain one another.

Takeaway: Maxwell’s equations describe how fields respond to charge and current and how changing electric and magnetic fields are linked.

How electromagnetic waves travel

In a region with no charge or current, Maxwell’s equations predict waves in both fields. In vacuum, the electric field obeys the wave equation

\nabla^2\mathbf{E}=\mu_0\varepsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2}.

Comparing this with the standard wave equation gives the speed of light in vacuum:

c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}\approx 3.00\times10^8\ \text{m/s}.

For any wave, speed, frequency, and wavelength are related by

c=f\lambda.

A plane is transverse: its electric field, magnetic field, and direction of travel are mutually perpendicular. The fields oscillate in phase, and their amplitudes are related by

E_0=cB_0.

For example, a wave traveling in the $+x$ direction can have its electric field along $+y$ and its magnetic field along $+z$ . The direction of travel follows the right-hand rule for $\mathbf{E}\times\mathbf{B}$ .

Electromagnetic waves need no material medium. They include radio waves, microwaves, visible light, and other forms of electromagnetic radiation. Accelerating charges, such as electrons oscillating in a transmitting antenna, can generate them.

Takeaway: In vacuum, an travels at $c$ , with perpendicular, in-phase electric and magnetic fields.

Energy carried by electromagnetic waves

Electromagnetic fields store energy. In vacuum, the energy per unit volume, called energy density, is

u=\frac{1}{2}\varepsilon_0 E^2+\frac{B^2}{2\mu_0}.

For a plane wave, the electric and magnetic contributions to the energy density are equal. Energy also moves with the wave. The gives the direction and rate of energy flow per unit area:

\mathbf{S}=\frac{1}{\mu_0}\mathbf{E}\times\mathbf{B}.

Its units are watts per square metre. For a uniform wave crossing an area perpendicular to its direction of travel, the power crossing that area is

P=SA.

For a sinusoidal wave, instruments usually measure the average over many cycles. This average energy flow per unit area is the :

I=\langle S\rangle=\frac{1}{2}\varepsilon_0 c E_0^2=\frac{cB_0^2}{2\mu_0}.

Because is proportional to the square of field amplitude, doubling either field amplitude makes the average four times greater.

Example: For a sinusoidal wave in vacuum with $I=100\ \text{W/m}^2$ , the electric-field amplitude is

E_0=\sqrt{\frac{2I}{\varepsilon_0c}}\approx275\ \text{V/m}.

The magnetic-field amplitude is

B_0=\frac{E_0}{c}\approx9.2\times10^{-7}\ \text{T}.

Takeaway: The describes electromagnetic energy flow, while gives its average magnitude per unit area.

Energy conservation and fields

Energy conservation for electromagnetic fields is summarized by Poynting’s theorem:

\frac{\partial u}{\partial t}+\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}.

The equation accounts for three things: change in field energy within a region, energy flowing out through the region’s boundary, and energy transferred between the field and matter. A decrease in field energy can correspond to energy leaving the region or being transferred to charges. The quantity $\mathbf{J}\cdot\mathbf{E}$ is the rate of energy transfer to matter per unit volume.

This connects the field description to the energy carried by a wave: energy can move through space as electromagnetic fields and can also be transferred to charged matter.

Takeaway: Field energy is conserved when energy flow and transfer to matter are included.