Quantum Mechanics - Comparing Rayleigh-Jeans formula with Planck's

[Cimster]

Homework Statement

Over what range in frequencies does the Rayleigh-Jeans formula give a result within 10% of the Planck blackbody spectrum?

Homework Equations

Planck Formula:

$p(v)dv = \frac{8πh}{c^3}(\frac{v^3}{e^(hv/kT)-1})dv$


Rayleigh-Jean Formula:

$p(v)dv = \frac{8πkT}{c^3}(v^2)dv$

The Attempt at a Solution

I've used the Rayleigh-Jean and Planck blackbody formulas, so I'm not unfamiliar with them. But I'm not even sure where to start with this question. The only two approaches I can think of are to start arbitrarily picking frequency values, solving both equations, and test for the percent errors... or to combine the two formulas along with the percentage error formula in a massively complex equation that I would need Mathematica to solve, which can't possibly be the correct way of going about it.

I have a feeling that I'm going to kick myself over this, but can someone provide me with some guidance on how to go about this?

 

[TSny]

Write an equation expressing that the difference between the two distributions is 10% of the Planck distribution. Simplify as far as possible.

For convenience let x = h$\nu$/kT and write the equation in terms of x.

I think you'll end up with an equation that cannot be solved exactly for x, as you suspected. See if you can think of a way to find a good approximate solution for x without resorting to Mathematica.