Poynting vector and intensity

An electromagnetic wave carries energy. The quantity that describes its flow is the Poynting vector:

Poynting vector

$\ev{\vec{I} = \frac{\vec{E}\times\vec{B}}{\mu_0}}$

Its direction is that of propagation of the wave (it coincides with $\vec{E}\times\vec{B}$), and its magnitude is measured in $\text{W}/\text{m}^2$: it is a power per unit area, i.e. the energy crossing, every second, a square metre placed perpendicular to the wave.

Key formula — Intensity

$|\vec{I}| = \frac{|\vec{E}|\,|\vec{B}|}{\mu_0}$ Time average of a harmonic wave: $\langle |\vec{I}|\rangle = \frac{E_0^2}{2\,c\,\mu_0}$

The factor $1/2$ in the average comes from the average of $\cos^2$ over a period (which equals $1/2$): the instantaneous intensity oscillates, but what a sensor measures is its mean value. Using $|B_0| = E_0/c$ the average can also be rewritten as $\langle I\rangle = \tfrac{1}{2}\varepsilon_0 c\,E_0^2$, a very convenient form for problems.

Since intensity is proportional to $E_0^2$, doubling the amplitude of the field quadruples the energy carried: it is the same quadratic dependence that governs all waves. For an isotropic point source, the intensity decreases as $1/r^2$ with distance, because the same power is spread over spheres of growing surface area $4\pi r^2$.

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Topics: Electromagnetic waves Concepts: Poynting vector · Intensity of a wave · Electromagnetic wave

Related exercises: Problem — Which EM wave · Worked exercise — Solar panel · Problem — WiFi intensity at 10 metres