Suppose a hot system at temperature T placed in an environment at a lower temperature T0. The system emits more radiation than it absorbs and cools down while the surrounding absorb radiation from the systems and warm up. The system is losing power by emitting radiations at a rate.
P1 = eAσT4
and is receiving energy by absorbing radiations at a rate
P2 = aAσT04
Here 'a' is a pure number between 0 and 1 showing the relative ability of the field to absorb radiation from its environment. Note that this 'a' is different from the absorptive energy 'a'. In thermal equilibrium, both the object and the environment have the similar temperature and,
P1 = P2 or eAσTc4 = aAσTc4
or e = a
Thus, when T > T0, the total rate of heat transferred from the body to the environment is,
dQ/dt = eAσ (T4 - T04) or mc(-dT/dt) = eAσ(T4 - T04)
Rate of cooling, (-dT/dt) = eAσ/mc(T4 - T04) or -dT/dt ∝ (T4 - T04)
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