Let's consider a very simple case of linear absorption. Sorry for ugly mathematical expressions.
Two counter propagating isotropic plane waves eikz and e-ikz meet in a layer of slightly absorbing medium and form a standing wave (SW). Let's assume the natural absorption coefficient alpha very small so that the SW's amplitude is ~2sin(kz). Thereby its intensity I = (2sin(kz))^2 = 2 * (1 - cos(2kz)), while intensity of each traveling wave is 1.
Now let's try to calculate absorbed power density (intensity) dP in a thin layer dz at one of the SW's maxima. It should be dP = alpha * I * dz = 4 * alpha * dz, isn't it? At the same time, each travelling wave loses alpha * dz when propagatig through dz, which is in sum 2 * alpha * dz -- two times less than dP. How to explain this difference?
Hello Anton, how are you? I think your aproximmation to ~2sin(kz) makes the difference.
Lets assume E1 = A e-alpha*z e-ikz and E2 = A ealpha*z eikz and ET = E1+E2 and then I = ET ET* (where * stands for conjugation), if you calculate dP in this case you'll have the answer.
I hope this help you.
Lucas.
Dear Lucas,
thank you for your responce!
But it doesn't help. I'll try to define the problem more general.
It is well known that in case of the interference of two beams intensity at a point z is not just sum of the intensities but
I = I1 + I2 + sqrt(2 I1 I2) cos(phi)
where phi is the phase difference changing with z and constant in time.
Therefore, if cos(phi) differs from zero there is a difference between (alpha * I * dz) and (alpha * (I1 + I2) * dz). I can't understand how local (in a small vicinity of z) energy balance is satisfied here.
.
going through the following paper might help
http://puhep1.princeton.edu/~kirkmcd/examples/destructive.pdf
.
The correct answer (see an elementary E&M book) e.g. Jackson or Griffiths is I*alpha*dz , or, better, assuming that only the dielectric constant is complex, the energy loss in a small volume dV is (1/2)dV Im(EE* \varepsilon) where the electric field satisfies E=Re(E exp(-i\omega t)) and \varepsilon is the dielectric constant at that frequency. I am not sure what problem you "see" with this result, but I think it has to have to do with other approximations you are making. The result I quote is well known and consistent with local energy balance and detailed analysis of the Poynting vector and Maxwell stress-energy tensor.
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