Rayleigh-Jeans Law

Question

My question is simple, why do we believe Rayleigh-Jeans law to be absurd? Is the Ultra-violet catastrophe incorrect or is it only because we can not create or know of a perfect emitter? I am a bit confused as to why Quanta was necessary.

Answer 1

The Rayleigh-Jeans law states that the spectral radiance of a black body at temperature $T$ is given as $$ B(\nu,T) = \frac{2k_B T}{c^2}\nu^2$$ The total radiance is then $$ B_\text{tot}(T) = \int_0^\infty B(\nu,T)\mathrm{d}\nu$$ but the integral $\int_0^\infty x^2\mathrm{d}x$ does not converge. Worse, it is infinte, i.e. $\lim_{k\to\infty}\int_0^k x^2\mathrm{d}x "=" \infty$.

Hence, the classically derived Rayleigh-Jeans law predicts that the radiance of a a blackbody is infinite. Since radiance is power per angle and unit area, this also implies that the total power and hence the energy a blackbody emitter gives off is infinite, which is patently absurd. This is called the ultraviolet catastrophe because the absurd prediction is caused by the classical law not predicting the behaviour at high frequencies/small wavelengths correctly.