Proving the Total Energy Density of Planck-Body Law: λ to f Domain Conversion

[thatguy14]

Homework Statement

Starting from the Planck-Body Law

I$_{λ}$dλ = $\frac{2\pi c^{2}h}{λ^{5}}$ $\frac{1}{e^{hc/(λkT)} - 1}$dλ

where λ is the wavelength, c is the speed of light in a vaccuum, T is the temperature, k is Boltzmann’s constant,
and h is Planck’s constant, prove that the total energy density over all wavelengths is given by

I$_{tot}$ = aT$^{4}$

and express a in terms of pi,k,h,c

Homework Equations

λ = c/f

The Attempt at a Solution

Our teacher gives us a hint "think about whether it is better to do the integral in the wavelength or frequency domain" - which in this case means he wants us to switch to the frequency domain. I did try a bunch of things but I am just not sure if my first step is correct. To switch to the frequency domain, all I havr to do is plug in

λ = c/f
and
dλ = -c/f$^{2}$

correct? Or is this first step wrong

 

[Simon Bridge]

dλ = -c/f² ... there should be a df in there somewhere.

$$\renewcommand{\d}{\;\text{d}}$$ $$I_\lambda \d\lambda = \frac{2\pi c^2 h}{\lambda^5}\frac{\d \lambda}{e^{hc/\lambda kT}-1}$$ ... can you see why it may be easier to change to frequency domain?

Note: when a hint says to "think about" something, you usually get extra marks for showing that you actually thought about it instead of just taking the hint blindly. Sometimes a teacher will hive you a false "think about" in the hint and you are supposed to dismiss it with reasoning. Therefore: check that the hint makes sense.