Approximations to Planck's law for the spectral emissive power are the Wien and RayleighJeans spectral distributions, which are useful for the extreme low and high limits of the product \(\lambda T\), respectively.

(a) 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 \(\lambda T=2898 \mu \mathrm{m} \cdot \mathrm{K}\). This form is known as Wien's law.

(b) 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 \(\lambda T=100,000 \mu \mathrm{m} \cdot \mathrm{K}\). This form is known as the Rayleigh-Jeans law.