Question

i) Write an expression for Planck’s law for energy density of photons in a cavity and
calculate total energy density, u. ii) Consider sun as a black body whose interior consists
of photons gas at T=3x 10^6 K Calculate the energy density of the solar radiations.
Take σ=7.56x 10^-16 Jm^-3 k^-4

Answer 1

50645, Physics, Molecular Physics — Thermodynamics

Question i) Write an expression for Plancks law for energy density of photons in a cavity and calculate total energy density, u. ii) Consider sun as a black body whose interior consists of photons gas at $T=3\cdot 10^{6}$ K Calculate the energy density of the solar radiations. Take $\sigma=7.56x10^{-16}Jm^{-3}k^{-4}$

Solution Planck’s law for energy density of photons is

$B_{\nu}(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{k_{\rm B}T}}-1}$

where $\nu$ is frequency of photons.

Total energy density is Planck’s law integrated over all frequencies:

$P=\int_{0}^{\infty}d\nu\int_{0}^{\pi/2}d\theta\int_{0}^{2\pi}d\phi\,B_{\nu}(T)\cos(\theta)\sin(\theta)=\sigma\,T^{4}$

Now we can find energy density of the solar radiation. Real value of Boltzman constant is $\sigma=5.67\cdot 10^{-8}Js^{-1}m^{-2}K^{-4}$. Hence

$P=\sigma T^{4}=5.67\cdot 10^{-8}\cdot 3\cdot 10^{6}\approx 0.17Js^{-1}m^{-2}$