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ck99
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Homework Statement
Prove that the Planck function increases monotonically with temperature.
Homework Equations
Bv(T) = 2hv3c-2(ehv/kT - 1)-1
The Attempt at a Solution
I first went through this piece-by-piece, but I am not a mathematician so I don't know if this constitutes "proof"!
1) First consider ehv/kT and note that this function will decrease as T increases.
2) This means (ehv/kT - 1) will also decrease.
3) Therefore (ehv/kT - 1)-1 will increase with T.
4) The other elements are independant of T, so the Planck function will increase with T.
I also thought of taking the derivative with respect to T of the Planck function, to see if the gradient ever reached zero to indicate a stationary point. After doing a couple of substitutions, I got
dBv(T)/dT = 2h2v4c-2k-1(e-hv/kT + 1)-1
I'm not sure that is correct, and I think it tells me that gradient gets smaller as T increases. This doesn't help my argument much!