I was just working on a problem that asked me to show that Plank's Law for black body radiation is approximately equal to Rayleigh-Jeans Law, which expresses the energy density of black body radiation as a function of wavelength. I was to show that this relation is only true at high wavelengths of light. To solve the problem, I expressed e in Plank's Law as a Taylor Polynomial with two terms, T2(x). Doing this resulted in me achieving the same expression as the Rayleigh-Jenes Law. My question is, why does what appears to be just making Plank's Law less accurate (expanding e as a Taylor Series) result in me proving the expressions are approximately equal at LONGER WAVELENGTHS?? Or in other words, why does making Plank's Law less accurate result in it being true for longer wavelengths? Thanks for any replies.
The Rayleigh-Jeans law was discovered first, before Planck's law. It is only valid for long wavelengths. It is constructed to be so. The Rayleigh-Jeans law does not work for short wavelengths (running into the so-called Ultra-violet catastrophe). Planck's law is true over all wavelengths. It's just that it roughly matches Rayleigh-Jeans law at long wavelengths because Planck's law is valid for long wavelengths just like the Rayleigh-Jeans law was constructed to be.
You said you Taylor expanded Planck's law in terms of x. Think carefully about what is x. A Taylor expansion is only accurate for small values of x. How does that relate to wavelength or frequency? There is more than one Taylor expansion possible.