## Physics Friday 136

Part 16: Energy in Electromagnetic Waves

As we saw here, the flow of energy in electromagnetic fields is given by the Poynting vector , the power flowing per unit of cross-sectional area. Now, let us consider electromagnetic plane waves in vacuum. As we showed here, they are transverse waves with the electric and magnetic fields both perpendicular to the direction of propagation, and that the electric and magnetic fields are perpendicular to each other as well, and proportional, with . Since the fields are perpendicular, the magnitude of their cross product is simply the product of their magnitudes:
, and we can see by the right-hand rule that S will be in the direction of propagation , so electromagnetic waves transport energy in their direction of propagation; an obvious result. We also see that since S is proportional to the square of a field magnitude, we recover the important optical property that the intensity of an eletromagnetic wave is proportional to the square of its amplitude.
More specifically, for a linearly-polarized wave, we have sinusoidally varying fields. With z the direction of propagation and polarization in the x direction, we then have

,
and so
.
The cosine-squared term oscillates between zero and one, and has a time average of 1/2. Taking the time average, then, we have
,
where Erms is the root-mean-square amplitude of the electric field, and which is equal to the peak amplitude over the square root of two.

For circularly-polarized waves, the magnitudes of the electric and magnetic fields are constant, so the Poynting vector is a constant; with propagation again in the z direction,
.
For general elliptical polarization, we can again express in terms of the root-mean-square amplitude (with mean over time) as
,
which reduces to the linear or circular case in each of those limits.

Also, since the energy density for electric fields is , and the energy density for magnetic fields is . Applying these to our wave, and taking the time average over a cycle, the average energy density is

And dividing the energy flux by the energy density confirms that the speed of the energy flow is
,
the speed of light in vacuum, as expected.