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

[phystudent17]

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.

[borgwal]

"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!

[phystudent17]

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>

[borgwal]

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.