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Homework Help: Relating time averaged energy density to the Poynting vector per unit solid angle

  1. Oct 20, 2008 #1
    1. The problem statement, all variables and given/known data
    Basically, the problem states that a cavity at temperature T is emitting EM waves isotropically in all directions (with frequency distribution of Planck's Law). If the time averaged density is <e>, find the value of d<S>/dw where w is the solid angle and the quantity is the effective poynting vector magnitude per unit solid angle. Hence I am to show the power per unit area that passes in one direction (i.e. into solid angle of 2 pi) through any plane within the cavity is dP/dA= (c/4)<e>/ Note that the unit system is Gaussian. Basically, I am stuck at the first part of the problem.

    2. Relevant equations

    Some equations that I know are <S>=c<e>, the total solid angle for a sphere is 4 pi.

    3. The attempt at a solution

    I have a feeling the solution is really simple but I cannot get into the physics of it. Is d<S>/dw just <S>/ 4pi= (c/4 pi)<e>? But then integrating over a solid angle of 2 pi gives me (c/2)<e> which is off by a factor of 2. And I really don't get the solid angle business. Can someone point me in the right direction? Thanks.
  2. jcsd
  3. Oct 20, 2008 #2
    "But then integrating over a solid angle of 2 pi gives me (c/2)<e> which is off by a factor of 2"

    That's because you should be integrating over a solid angle of 4 pi, as you already know!
  4. Oct 20, 2008 #3
    but now i want the power per unit area passing through one direction and that has a solid angle of 2 pi not 4 pi. the qn requires me to show that integrating over the solid angle of 2 pi gives me (c/4)<e>
  5. Oct 20, 2008 #4
    I did misread your question, sorry about that. In any case, the relation |<\vec{S}>|=c<e> holds for plane waves propagating in a given direction. It's not a general relation.
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