**ELECTROMAGNETIC ENERGY FLOW AND THE POYNTING VECTOR**

Electromagnetic waves such as those we have described are traveling waves that transport energy from one region to another. For instance, in the wave described in Section 33-3 the E and i fields advance with time into regions where originally no fields were present and carry the energy density u with them as they advance. We can describe this energy transfer in terms of energy transferred per unit time per unit cross-section area, or power per unit area, for an area perpendicular to the direction of wave travel. To see how the energy flow is related to the fields, consider a stationary plane, perpendicular to the x-axis, that coincides with the wave front at a certain time. In a time dt

after this, the wave front moves a distance dx =c dt to the right of the plane. Considering an area A on this stationary plane (Fig. 33-11), we note that the energy in the space to the right of this area must have passed through the area to reach the new location. The volume dV of the relevant region is the base area A times the length c dt, and the energy dU in this region is the energy density u times this volume:

The derivation of Eq. (33-24) from Eq. (33-23) is left as a problem (see Exercise 33-9). The units of S are energy per unit time per unit area, or power per unit area. The SI unit of S is 1Jls • m2or 1W/m2:

We can define a vector quantity that describes both the magnitude and direction of the energy flow rate:

The vector Sis called the Poynting vector; it was introduced by the British physicist John Poynting (1852-1914). Its direction is in the direction of propagation of the wave. Since E are if are perpendicular, the magnitude of Sis given by S = Eq. (33-24) this is the flow of energy per unit area and per unit time through a cross-section area perpendicular to the propagation direction. The total energy flow per unit time sinusoidal waves studied in Section 33-4 as well as for other more complex electric and magnetic fields at any point vary with time, so the Poynting vec- 2!Il)’ point is also a function of time. Because the frequencies of typical ,”;;”–,….,.””meticwaves are very high, the time variation of the Poynting vector is so rapid . most appropriate to look at its average value. The average value of the magnify Sat a point is called the intensity of the radiation at that point. The SI unit of ty is the same as for S, 1W/m2 (watt per square meter). Let’s work out the intensity of the sinusoidal wave described by Eqs. (33-16). We substitute these expressions into Eq. (33-24): For a wave traveling in the -x-direction, represented by Eqs. (33-19), the Poynting vector is in the -x-direction at every point, but its magnitude is the same as for a wave traveling in the +x-direction. Verifying these statements is left to you (see Exercise 33-12). CAUTION ~ At any point x, the magnitude of the Poynting vector varies with time. Hence the instantaneous rate at which electromagnetic energy in a sinusoidal plane wave arrives at a surface is not constant. This may seem to contradict everyday experience; the light from the sun, a light bulb, or the laser in a grocery-store scanner appears steady and unvarying in strength. In fact the Poynting vector from these sources does vary in time, but the variation isn’t noticeable because the scillation frequency is so high (around 5 x 1014Hz for visible light). All that you sense is the average rate at which energy reaches your eye, which is why we commonly use intensity (the average value of S) to describe the strength of electromagnetic radiation.