Prove that Planck function increases with temperature

[ck99]

Homework Statement

Prove that the Planck function increases monotonically with temperature.

Homework Equations

B_v(T) = 2hv³c^-2(e^hv/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 e^hv/kT and note that this function will decrease as T increases.

2) This means (e^hv/kT - 1) will also decrease.

3) Therefore (e^hv/kT - 1)^-1 will increase with T.

4) The other elements are independent 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

dB_v(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!

 

[TSny]

Homework Statement

Prove that the Planck function increases monotonically with temperature.

Homework Equations

B_v(T) = 2hv³c^-2(e^hv/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 e^hv/kT and note that this function will decrease as T increases.

2) This means (e^hv/kT - 1) will also decrease.

3) Therefore (e^hv/kT - 1)^-1 will increase with T.

4) The other elements are independent of T, so the Planck function will increase with T.

The above argument sounds good to me.

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

dB_v(T)/dT = 2h²v⁴c^-2k^-1(e^-hv/kT + 1)^-1

That's not the correct result for the derivative. Ignoring the constants out front, you have

B = (x-1)^-1 where x = e^hv/kT.

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

 

[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.