Hello, The energy density of an electromagnetic wave is [itex]ε_{0}E^{2}[/itex]. To calculate the energy flux, at least in the derivation's I've seen, people just multiply by the speed of the wave, i.e., c. But doesn't this assume that the energy density is constant at all points?; but E changes periodically! Why isn't it then the integral of the energy density in the corresponding volume, so it would give something close to a half of the usual answer i see!? Thanks in advance
Just read in here: http://hep.ph.liv.ac.uk/~hutchcroft/Phys258/CN6EMWaves.pdf that I was right yeah. Walter Lewin's lecture was a bit missleading.
One has to be very careful with terminology and keep in mind exactly which quantity is being discussed. The instantaneous power density (W/m^{2}) passing through a surface oriented perpendicular to the wave propagation direction is indeed ε_{0}cE^{2}. However, the magnitude of E oscillates rapidly between 0 and the amplitude E_{max} (often called E_{0}). Finding the time-average over a whole number of cycles gives half of the maximum power density, so the time-averaged power density (which is what we can actually measure in practice) is (1/2)ε_{0}cE_{max}^{2} which is often written as (1/2)ε_{0}cE_{0}^{2}. I haven't watched Levin's lecture so I don't know which terminology he's using.