Question

Write an expression for Plank's law of radiation

Answer 1

Plank's law state that the spectral radiance ${{B}_{\nu }}$ of a physical body for frequency $\nu$  at absolute temperature T is given by

$${{B}_{\nu }}\left( \nu ,T \right)=\frac{2h{{\nu }^{3}}}{{{c}^{2}}}\frac{1}{{{e}^{\frac{h\nu }{kT}}}-1}$$

where k is the Boltzmann constant, h is the Planck constant, and c  is the speed of light in the medium. Spectral radiance ${{B}_{\nu }}$ is the power, emitted per unit area of the body, per unit solid angle of emission, per unit frequency.

Considering that

$$\nu =\frac{c}{\lambda }$$

where λ is the radiation wavelength, the spectral radiance can also be expressed per unit wavelength λ instead of per unit frequency, that is

$${{B}_{\lambda }}\left( \lambda ,T \right)=\frac{2h{{c}^{2}}}{{{\lambda }^{5}}}\frac{1}{{{e}^{\frac{hc}{\lambda kT}}}-1}$$

There is a relationship between these two forms of Plank's law

$${{B}_{\lambda }}\left( \lambda ,T \right)d\lambda =-{{B}_{\nu }}\left( \nu ,T \right)d\nu$$

where $d\nu =-\frac{c}{{{\lambda }^{2}}}d\lambda$  holds true.