6.0 Stefan-Boltzman law

The rate at which a body radiate energy is proportional to the fourth power of its absolute temperature and is known as Stefan's law. $$u = \sigma A{T^4}$$ where

$u$ is power in $Watts\ (J/s)$ radiated by the body,

$A$ is the surface area in ${m^2}$,

$\sigma $ is a universal constant called stefan's constant,

$\sigma = 5.67 \times {10^{ - 8}}\,W/{m^2} - {K^4}$.

A body which is not a perfect black body, emits less energy.

However, it is proportional to fourth power of %T% and can be written as $$u = \sigma eA{T^4}$$ where $e$ is a fraction between $0$ and $1$ and is known as the emissivity of the body.

It is unity for a perfect black body and zero for completely reflecting surface.

From kirchhoff's law,$$\begin{equation} \begin{aligned} \frac{{{E_{body}}}}{{{E_{perfect\,black\,body}}}} = a \\ \frac{{e\sigma A{T^4}}}{{\sigma A{T^4}}} = a \\ \Rightarrow a = e \\\end{aligned} \end{equation} $$

Consider a body at the temperature $T$ and the surroundings at ${T_0}$, we can write, the energy absorbed by the body per unit time as $${u_0} = e\sigma A{{T_0}^4}$$ The energy emitted by the body per unit time is$$u = e\sigma A{T^4}$$Thus, the net loss of thermal energy per unit time can be written as$$\begin{equation} \begin{aligned} \Delta u = u - {u_0} \\ = e\sigma A\left( {{T^4} - {T^4}_0} \right) \\\end{aligned} \end{equation} $$