Chapter 12, Problem 12.27P

Approximations to Planck’s law for the spectral emissive power are the Wien and Rayleigh-Jeans spectral distributions, which are useful for the extreme low and high limits of the product $\lambda T$ , respectively.

1. Show that the Planck distribution will have the form
$E_{\lambda ,b}(\lambda ,T)\approx \frac{C_{`1}}{{\lambda }^5}\exp \left(-\frac{C_2}{\lambda T}\right)$
when $C_2/\lambda T\gg 1$ and determine the error (compared to the exact distribution) for the condition $\lambda T=2898\mu m\cdot K$ This form is known as Wien’s law.

2. Show that the Planck distribution will have the form
$E_{\lambda ,b}(\lambda ,T)\approx \frac{C_{`1}}{C_2}\frac{T}{{\lambda }^4}$
when $C_2/\lambda ,T\ll 1$ and determine the error compared to the exact distribution) for the condition $\lambda T=100,000\mu m\cdot K$ . This form is known as the Rayleigh-Jeans law.