Stephan-Boltzmann law, solid angle integral, and an extra pi factor

[ck99]

Homework Statement

This isn't a homework question, I am writing up my scribbled notes from todays lecture and have got stuck on some calculus, and lost the thread of the argument. Last week, we integrated Plancks law to find

B(T) = ∫ B_v(T) dv

= 2∏⁴(kT)⁴ / 15c²h³

Then defined the Stephan Boltzmann constant σ as

σ = 2∏⁵k⁴ / 15c²h³


To get the Stephan Boltzmann Law expressed as

B(T) = σT⁴ / ∏


Today, we start with the Stephan-Boltzmann law expressed as F = σT⁴ and start to investigate where our previous π^-1 term comes from.

We start by considering an area element in a flux, with area dA and unit vector n. We define flux passing through this element as

(1) dF = dΩ cosθ B(T)

Integrate to get

(2) F = B(T) 2∏ ∫sinθ cos θ dθ (integral from 0 to pi/2)

(3) F = B(T) 2∏ ∫μ dμ (integral from 0 to 1)

(4) F = σT⁴

Homework Equations

All stated above (I hope!)

The Attempt at a Solution

My questions are, firstly, are F and B(T) both fluxes? I understand how we got to defining B(T) but I'm not exactly sure what it is!

Second, what does the cos θ term in eqn 1 describe?

Third, how do we get from eqn 2 to eqn 3, and how is the integral of μ equal to 1/2 in order for eqn 4 to be true?

For the first question, I have found that B is used for magnetic flux, but I don't think that is what is being described here.

For the second, I think this is to do with the angle of emission compared to the normal vector n we defined, so it would be necessary to integrate over all possible values of θ, but surely this would be a solid angle term like Ω? Or is that where the dΩ term comes from? Or is dΩ an integral over the sum of the surface elements?

For the third, I am totally stuck. Someone else asked for help on this, and he wrote down the following:

∫sin θ f(sin θ cos θ) dθ (integral from 0 to pi) = ∫ f{√(1 - μ²), μ} dμ (integral from -1 to 1)

dθ sin θ = -d cos θ = -dμ

That does not help me in the slightest, in fact I am more confused than ever now I have typed it all out! I have spent over an hour trying to figure out what is going on with this integration, my calculus is extremely poor (I seem to forget it faster than I learn it) so the explanations/lecture notes I have found online are not very helpful to a dunce such as myself. Could someone point me towards an idiots guide to whatever the heck this is?

 

[_coyote_]

Or lay out in very simple terms what is going on, so I can actually understand it this time? I am able to do the integration myself, I just need help understanding what is going on and why each step is necessary.Thank you in advance for any help you can offer!