Rayleigh–Jeans law

Comparison to Planck's law

In 1900 Max Planck empirically obtained an expression for black-body radiation expressed in terms of wavelength λ = c/ν (Planck's law): $${\displaystyle B_{\lambda }(T)=\frac{2h{c}^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda k_{\text{B}}T}}-1},}$$ where h is the Planck constant, and k_B is the Boltzmann constant. Planck's law does not suffer from an ultraviolet catastrophe and agrees well with the experimental data, but its full significance (which ultimately led to quantum theory) was only appreciated several years later. Since $${\displaystyle e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots ,}$$ then in the limit of high temperatures or long wavelengths, the term in the exponential becomes small, and the exponential is well approximated with the Taylor polynomial's first-order term: $${\displaystyle e^{\frac{hc}{\lambda k_{\text{B}}T}}\approx 1+\frac{hc}{\lambda k_{\text{B}}T}.}$$

So $${\displaystyle \frac{1}{e^{\frac{hc}{\lambda k_{\text{B}}T}}-1}\approx \frac{1}{\frac{hc}{\lambda k_{\text{B}}T}}=\frac{\lambda k_{\text{B}}T}{hc}.}$$

This results in Planck's blackbody formula reducing to $${\displaystyle B_{\lambda }(T)=\frac{2c{k}_{\text{B}}T}{{\lambda }^4},}$$ which is identical to the classically derived Rayleigh–Jeans expression.

The same argument can be applied to the blackbody radiation expressed in terms of frequency ν = c/λ. In the limit of small frequencies, that is ${\displaystyle h\nu \ll k_{\text{B}}T}$, $${\displaystyle B_{\nu }(T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{k_{\text{B}}T}}-1}\approx \frac{2h{\nu }^3}{c^2}\cdot \frac{k_{\text{B}}T}{h\nu }=\frac{2{\nu }^2{k}_{\mathrm{B}}T}{c^2}.}$$

This last expression is the Rayleigh–Jeans law in the limit of small frequencies.

Consistency of frequency- and wavelength-dependent expressions

When comparing the frequency- and wavelength-dependent expressions of the Rayleigh–Jeans law, it is important to remember that $${\displaystyle \frac{dP}{d\lambda }=B_{\lambda }(T)}$$ and $${\displaystyle \frac{dP}{d\nu }=B_{\nu }(T).}$$ Note that these two expressions then have different units, as a step ${\displaystyle d\lambda }$ in wavelength is not equivalent to a step ${\displaystyle d\nu }$ in frequency. Therefore, $${\displaystyle B_{\lambda }(T)\ne B_{\nu }(T),}$$ even after substituting the value ${\displaystyle \lambda =c/\nu }$, because ${\displaystyle B_{\lambda }(T)}$ has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit wavelength, whereas ${\displaystyle B_{\nu }(T)}$ has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit frequency. To be consistent, we must use the equality $${\displaystyle B_{\lambda }d\lambda =dP=B_{\nu }d\nu ,}$$ where both sides now have units of power (energy emitted per unit time) per unit area of emitting surface, per unit solid angle.

Starting with the Rayleigh–Jeans law in terms of wavelength, we get $${\displaystyle B_{\lambda }(T)=B_{\nu }(T)\frac{d\nu }{d\lambda },}$$ where $${\displaystyle \frac{d\nu }{d\lambda }=\frac{d}{d\lambda }\left(\frac{c}{\lambda }\right)=-\frac{c}{{\lambda }^2}.}$$ This leads to $${\displaystyle B_{\lambda }(T)=\frac{2k_{\text{B}}T{\left(\frac{c}{\lambda }\right)}^2}{c^2}\times \frac{c}{{\lambda }^2}=\frac{2c{k}_{\text{B}}T}{{\lambda }^4}.}$$

Other forms of Rayleigh–Jeans law

Depending on the application, the Planck function can be expressed in 3 different forms. The first involves energy emitted per unit time per unit area of emitting surface, per unit solid angle, per spectral unit. In this form, the Planck function and associated Rayleigh–Jeans limits are given by $${\displaystyle B_{\lambda }(T)=\frac{2h{c}^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda k_{\text{B}}T}}-1}\approx \frac{2c{k}_{\text{B}}T}{{\lambda }^4}}$$ or $${\displaystyle B_{\nu }(T)=\frac{2h{\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{k_{\text{B}}T}}-1}\approx \frac{2k_{\text{B}}T{\nu }^2}{c^2}.}$$

Alternatively, Planck's law can be written as an expression ${\displaystyle I(\nu ,T)=\pi B_{\nu }(T)}$ for emitted power integrated over all solid angles. In this form, the Planck function and associated Rayleigh–Jeans limits are given by $${\displaystyle I(\lambda ,T)=\frac{2\pi h{c}^2}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda k_{\text{B}}T}}-1}\approx \frac{2\pi c{k}_{\text{B}}T}{{\lambda }^4}}$$ or $${\displaystyle I(\nu ,T)=\frac{2\pi h{\nu }^3}{c^2}\frac{1}{e^{\frac{h\nu }{k_{\text{B}}T}}-1}\approx \frac{2\pi k_{\text{B}}T{\nu }^2}{c^2}.}$$

In other cases, Planck's law is written as $u(\nu ,T)=\frac{4\pi }{c}{B}_{\nu }(T)$ for energy per unit volume (energy density). In this form, the Planck function and associated Rayleigh–Jeans limits are given by $${\displaystyle u(\lambda ,T)=\frac{8\pi hc}{{\lambda }^5}\frac{1}{e^{\frac{hc}{\lambda k_{\text{B}}T}}-1}\approx \frac{8\pi k_{\text{B}}T}{{\lambda }^4}}$$ or $${\displaystyle u(\nu ,T)=\frac{8\pi h{\nu }^3}{c^3}\frac{1}{e^{\frac{h\nu }{k_{\text{B}}T}}-1}\approx \frac{8\pi k_{\text{B}}T{\nu }^2}{c^3}.}$$