# Prove that Planck function increases with temperature

1. Apr 13, 2013

### ck99

1. The problem statement, all variables and given/known data

Prove that the Planck function increases monotonically with temperature.

2. Relevant equations

Bv(T) = 2hv3c-2(ehv/kT - 1)-1

3. 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!
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Apr 13, 2013

### TSny

The above argument sounds good to me.
That's not the correct result for the derivative. Ignoring the constants out front, you have

B = (x-1)-1 where x = ehv/kT.

Try using the chain rule: dB/dT = (dB/dx)$\cdot$(dx/dT)

3. Apr 14, 2013

### ck99

Thanks TSny, I managed to compute the derivative correctly and that can also be put in terms of positive, increasing functions of T so I am sure that is adequate proof.