- #1
Airton Rampim
- 6
- 1
How can I find the relation between the radiance and the energy density of a black body? According to Planck's law, the energy density inside a blackbody cavity for modes with frequency ##\nu \in [\nu, \nu + \mathrm{d}\nu]## is given by $$ \rho(\nu, T)\mathrm{d}\nu = \frac{8\pi\nu^2}{c^3}\frac{h\nu}{\exp\left(\frac{h\nu}{k T}\right) - 1}\mathrm{d}\nu. $$ Integrating over all the frequencies we will have
$$ \rho_T = \intop_0^{\infty}\rho(\nu, T)\mathrm{d}\nu = \frac{8\pi h}{c^3} \intop_0^\infty \frac{\nu^3}{\exp\left(\frac{h\nu}{k T}\right) - 1}\mathrm{d} \nu $$
$$ \rho_T = \frac{8\pi h}{c^3}\left(\frac{k T}{h}\right)^4 \underbrace{\intop_0^\infty \frac{u^3}{\exp\left(u\right) - 1}\mathrm{d} u}_{\zeta(4)\Gamma(4) = \pi^4/15} = \frac{8\pi^5 k^4}{15 h^3 c^3}T^4 $$
At equilibrium, these modes will have the same energy of the blackbody that is proportional to the fourth power of the temperature. This is the same result predicted by Stefan-Boltzmann law for the radiance of a black body
$$ R_T = \intop_0^\infty R(\nu, T)\mathrm{d}\nu = \sigma T^4, $$
where ##R(\nu, T)\mathrm{d}\nu## is the radiance emitted by frequencies between ##\nu## and ##\nu + \mathrm{d}\nu## and ##\sigma## is the Stefan-Boltzmann constant. My question is how can I find the following relation
$$ R(\nu, T)\mathrm{d}\nu = \frac{c}{4} \rho(\nu, T)\mathrm{d}\nu? $$
$$ \rho_T = \intop_0^{\infty}\rho(\nu, T)\mathrm{d}\nu = \frac{8\pi h}{c^3} \intop_0^\infty \frac{\nu^3}{\exp\left(\frac{h\nu}{k T}\right) - 1}\mathrm{d} \nu $$
$$ \rho_T = \frac{8\pi h}{c^3}\left(\frac{k T}{h}\right)^4 \underbrace{\intop_0^\infty \frac{u^3}{\exp\left(u\right) - 1}\mathrm{d} u}_{\zeta(4)\Gamma(4) = \pi^4/15} = \frac{8\pi^5 k^4}{15 h^3 c^3}T^4 $$
At equilibrium, these modes will have the same energy of the blackbody that is proportional to the fourth power of the temperature. This is the same result predicted by Stefan-Boltzmann law for the radiance of a black body
$$ R_T = \intop_0^\infty R(\nu, T)\mathrm{d}\nu = \sigma T^4, $$
where ##R(\nu, T)\mathrm{d}\nu## is the radiance emitted by frequencies between ##\nu## and ##\nu + \mathrm{d}\nu## and ##\sigma## is the Stefan-Boltzmann constant. My question is how can I find the following relation
$$ R(\nu, T)\mathrm{d}\nu = \frac{c}{4} \rho(\nu, T)\mathrm{d}\nu? $$