Chapter 9, Problem 9.25E

The slope of the plot of energy versus wavelength for the Rayleigh-Jeans law is given by a rearrangement of equation $9.20$:

$\frac{d\rho }{d\lambda }=\frac{8\pi kT}{{\lambda }^4}$

What are the value and units of this slope for a blackbody having the following temperatures and at the following wavelengths? (a) $1000\text{K},500\text{nm}$; (b) $2000\text{K},500\text{nm}$; (c) $2000\text{K},5000\text{nm}$; (d) $2000\text{K},10,000\text{nm}$. Do the answers indicate the presence of an ultraviolet catastrophe?