Energy distribution laws of black body radiation

1.) Wein’s laws of Energy distributions→

A.) Wein's Fifth Power law→ The total amount of the energy emitted by a black body per unit volume at an absolute temperature in the wavelength range $\lambda$ and $\lambda + d\lambda$ is given as

$E\lambda \cdot d\lambda= \frac{A}{\lambda^{5}}f\left ( \lambda T \right ) \cdot d\lambda \qquad (1)$

Where $A$ is a constant and $f(\lambda T)$ is a function of the product $\lambda T$ and is given as

$ f\left ( \lambda T\right )=e^-\frac{hc}{\lambda kT}\qquad (2)$

From equation $(1)$ and $(2)$

$E_\lambda \cdot d\lambda = \frac{A}{\lambda ^{5}}e^\frac{-hc}{\lambda kT} \cdot d\lambda$

$E_\lambda \cdot d \lambda = A \lambda ^{-5} e^\frac{-hc}{\lambda kT} \cdot d \lambda$

Wien’s law energy distribution explains the energy distribution at the short wavelength at higher temperatures and fails for long wavelengths.

B.) Wein's Displacement law→ As the temperature of the body is raised the maximum energy shift toward the shorter wavelength i.e.

$\lambda_{m} \times T = Constant $

Where
$\lambda_m$- Wavelength at which the energy is maximum
$T$-Absolute temperature

Thus, if radiation of a particular wavelength at a certain temperature is adiabatically altered to another wavelength then temperature changes in the inverse ratio.

2.) Rayleigh-Jean’s law→ The total amount of energy emitted by a black body per unit volume at an absolute temperature T in the wavelength range $\lambda $ and $\lambda +d\lambda $ is given as

$E_{\lambda}.d\lambda = \frac{8\pi kt}{\lambda ^{4}}.d\lambda$

Where K– Boltzmann’s Constant which has valve $ 1.381\times 10^{23}\frac{J}{K}$

This law, explains the energy distribution at the longer wavelength at all temperatures and fails totally for the shorter wavelength.

Note→ The energy distribution curves of the black body show a peak while going towards the ultraviolet wavelength (shorter $ \lambda $) and then fall while Rayleigh-Jeans law indicates continuous rise only. This is the failure of classical physics.

3.) Stefan-Boltzmann Law→ The total amount of heat radiated by a perfectly black body per unit area per second is directly proportional to the fourth power of its absolute temperature $(T)$. i.e.

$E \propto T^{4}$

$E = \sigma T^{4}$

Where $\sigma$= Stefan’s Constant which has value $5.67\times 10^{-8} W-\frac{K^{4}}{m^{2}}$

It is a black body at absolute temperature $T$ is surrounded by another black body at absolute temperature $T_{0}$, The net amount of heat $E$ lost by the former per second per $cm^{2}$ is→

$E=\sigma (T^{4}-T_{0}^{^{4}})$