Electromagnetic fields carry energy through space. The local direction and rate of electromagnetic energy flow are described by the Poynting vector

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

Its direction is perpendicular to both $\mathbf E$ and $\mathbf B$, following the right-hand rule for the cross product. Its SI units are watts per square metre, so $\mathbf S$ is an electromagnetic energy flux: power crossing unit area.

Plane electromagnetic waves

For a plane wave in vacuum, $\mathbf E$, $\mathbf B$ and the propagation direction are mutually perpendicular, and

$$E=cB.$$

Therefore the Poynting vector points in the direction the wave propagates. Its instantaneous magnitude is

$$S=\frac{EB}{\mu_0}.$$

Using $B=E/c$ and $c^2=1/(\mu_0\varepsilon_0)$,

$$\boxed{S=\varepsilon_0cE^2=\frac{c}{\mu_0}B^2}.$$

For a sinusoidal plane wave

$$E(t)=E_0\cos(\omega t-kx),$$

the energy flux oscillates because it is proportional to $E^2$. The time-averaged intensity is

$$\boxed{I=\langle S\rangle=\frac12\varepsilon_0cE_0^2}$$

or equivalently

$$\boxed{I=\frac{c}{2\mu_0}B_0^2}.$$

Thus electromagnetic-wave intensity is proportional to the square of the field amplitude.

Worked example

A sinusoidal electromagnetic wave in vacuum has electric-field amplitude

$$E_0=100,\mathrm{V/m}.$$

Using

$$\varepsilon_0\approx8.85\times10^{-12},\mathrm{F/m},\qquad c\approx3.00\times10^8,\mathrm{m/s},$$

the average intensity is

$$I=\frac12(8.85\times10^{-12})(3.00\times10^8)(100)^2.$$

Therefore

$$\boxed{I\approx13.3,\mathrm{W/m^2}}.$$

The magnetic-field amplitude is

$$B_0=\frac{E_0}{c}\approx3.33\times10^{-7},\mathrm T.$$

Energy crossing a surface

If $\mathbf S$ is uniform over a flat surface of area $A$ whose normal makes angle $\theta$ with $\mathbf S$, then the instantaneous electromagnetic power crossing it is

$$P=SA\cos\theta.$$

For perpendicular incidence, this reduces to $P=SA$.

The Poynting vector is therefore the electromagnetic version of the general idea of wave intensity, but with directional information included. A more advanced local conservation law—the Poynting theorem—relates the divergence of this energy flux to changes in stored field energy and work done on matter.