# 16.3 Energy carried by electromagnetic waves  (Page 2/5)

 Page 2 / 5

The energy passing through area A in time $\text{Δ}t$ is

$u\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{volume}=uAc\text{Δ}t.$

The energy per unit area per unit time passing through a plane perpendicular to the wave, called the energy flux and denoted by S , can be calculated by dividing the energy by the area A and the time interval $\text{Δ}t$ .

$S=\frac{\text{Energy passing area}\phantom{\rule{0.2em}{0ex}}A\phantom{\rule{0.2em}{0ex}}\text{in time}\phantom{\rule{0.2em}{0ex}}\text{Δ}t}{A\text{Δ}t}=uc={\epsilon }_{0}c{E}^{2}=\frac{1}{{\mu }_{0}}EB.$

More generally, the flux of energy through any surface also depends on the orientation of the surface. To take the direction into account, we introduce a vector $\stackrel{\to }{S}$ , called the Poynting vector    , with the following definition:

$\stackrel{\to }{S}=\frac{1}{{\mu }_{0}}\stackrel{\to }{E}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}.$

The cross-product of $\stackrel{\to }{E}$ and $\stackrel{\to }{B}$ points in the direction perpendicular to both vectors. To confirm that the direction of $\stackrel{\to }{S}$ is that of wave propagation, and not its negative, return to [link] . Note that Lenz’s and Faraday’s laws imply that when the magnetic field shown is increasing in time, the electric field is greater at x than at $x+\text{Δ}x$ . The electric field is decreasing with increasing x at the given time and location. The proportionality between electric and magnetic fields requires the electric field to increase in time along with the magnetic field. This is possible only if the wave is propagating to the right in the diagram, in which case, the relative orientations show that $\stackrel{\to }{S}=\frac{1}{{\mu }_{0}}\stackrel{\to }{E}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{B}$ is specifically in the direction of propagation of the electromagnetic wave.

The energy flux at any place also varies in time, as can be seen by substituting u from [link] into [link] .

$S\left(x,t\right)=c{\epsilon }_{0}{E}_{0}^{2}{\text{cos}}^{2}\left(kx-\omega t\right)$

Because the frequency of visible light is very high, of the order of ${10}^{14}\phantom{\rule{0.2em}{0ex}}\text{Hz,}$ the energy flux for visible light through any area is an extremely rapidly varying quantity. Most measuring devices, including our eyes, detect only an average over many cycles. The time average of the energy flux is the intensity I of the electromagnetic wave and is the power per unit area. It can be expressed by averaging the cosine function in [link] over one complete cycle, which is the same as time-averaging over many cycles (here, T is one period):

$I={S}_{\text{avg}}=c{\epsilon }_{0}{E}_{0}^{2}\frac{1}{T}\underset{0}{\overset{T}{\int }}{\text{cos}}^{2}\left(2\pi \frac{t}{T}\right)dt.$

We can either evaluate the integral, or else note that because the sine and cosine differ merely in phase, the average over a complete cycle for ${\text{cos}}^{2}\left(\xi \right)$ is the same as for ${\text{sin}}^{2}\left(\xi \right)$ , to obtain

$⟨{\text{cos}}^{2}\xi ⟩=\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\left[⟨{\text{cos}}^{2}\xi ⟩+⟨{\text{sin}}^{2}\xi ⟩\right]=\frac{1}{2}⟨1⟩=\frac{1}{2}.$

where the angle brackets $⟨\text{⋯}⟩$ stand for the time-averaging operation. The intensity of light moving at speed c in vacuum is then found to be

$I={S}_{\text{avg}}=\frac{1}{2}c{\epsilon }_{0}{E}_{0}^{2}$

in terms of the maximum electric field strength ${E}_{0},$ which is also the electric field amplitude. Algebraic manipulation produces the relationship

$I=\frac{c{B}_{0}^{2}}{2{\mu }_{0}}$

where ${B}_{0}$ is the magnetic field amplitude, which is the same as the maximum magnetic field strength. One more expression for ${I}_{\text{avg}}$ in terms of both electric and magnetic field strengths is useful. Substituting the fact that $c{B}_{0}={E}_{0},$ the previous expression becomes

$I=\frac{{E}_{0}{B}_{0}}{2{\mu }_{0}}.$

We can use whichever of the three preceding equations is most convenient, because the three equations are really just different versions of the same result: The energy in a wave is related to amplitude squared. Furthermore, because these equations are based on the assumption that the electromagnetic waves are sinusoidal, the peak intensity is twice the average intensity; that is, ${I}_{0}=2I.$

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