Newton's law of cooling is a special case of
Wien's displacement law
From Stefan-Boltzman's law, the energy of the thermal radiation emitted per unit time by a blackbody of surface area A is given by, `u = σAT^4`
Where `σ` is Stefan's constant.
Suppose a body at temperature T is kept in a room at temperature T0.
According to Stefan's law, energy of the thermal radiation emitted by the body per unit time is `u = eσAT^4`
Here, e is the emissivity of the body.
The energy absorbed per unit time by the body is (due to the radiation emitted by the walls of the room
`Δu = eσAT_0^4`
Thus, the net loss of thermal energy per unit time is
`Δu = eσA ( T^4 - T_0^4 )`
Newton law of cooling is given by
`(dT)/(dt) = -bA( T - T_0 )`
This can be obtained from equation (i) by considering the temperature difference to be small and doing the binomial expansion.