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Homework Help: A little help with Poynting's theorem interpretation?

  1. Jan 6, 2010 #1
    I am doing some exam preparation and I am having trouble understanding what Poynting's theorem is all about. After much messing around with Maxwell's equations, it turns out that

    [tex] \frac{\partial U}{\partial t} + \nabla \cdot \vec{S} + \vec{E} \cdot \vec{J} = 0 [/tex] where the Poynting vector [tex] \vec{S} = \frac{1}{\mu_0} \vec{E}x\vec{B}[/tex] and the total energy density of the field [tex] U = \frac{\epsilon E^2}{2} + \frac{B^2}{2\mu_0}[/tex].

    One thing that is bothering me is that this is dealing with electromagnetic waves, right?
    So we have electric and magnetic sinusoidal waves oscillating in planes perpendicular to each other, and this electromagnetic wave is propagating in the direction of the vector S. I was thinking that surely the divergence of S should be equal to zero, since photons do not "diminish" as they travel through space, they just keep going and going, or we would not be able to see the stars etc.

    However, this equation is clearly saying that the divergence of S is not zero, so I must be interpreting this term wrong. What is the meaning of this term?
    Actually I have the same problem with the dU/dt term - why would this so-called energy density change with respect to time, wouldn't one expect this to be zero too?

    Also, I did not know what to make of the E and J scalar-product term at all. Can somebody please set me straight on this stuff?

    Many thanks.
  2. jcsd
  3. Jan 6, 2010 #2


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    One thing to keep in mind is that you're describing an electromagnetic wave in a vacuum. There are no sources so the current density [tex]\vec{J}[/tex] is zero. Also, [tex]\vec{S}[/tex] and [tex]U[/tex] do vary with time and space as the wave propagates because the electric and magnetic fields vary. However, they probably average to zero over time (for a given point in space).

    The [tex]\vec{E}\cdot\vec{J}[/tex] term corresponds to the work done on any charges present, like if an electromagnetic wave were propagating through a plasma.
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