# How are Stefan's law and Newton's law of cooling related?

Oct 14, 2015

Newton's law of cooling is a consequence of Stefan's law.

#### Explanation:

Let $T$ and $T '$ be the temperature of the body and the surroundings.
Then by Stefan;s law rate of heat loss of body is given by,
$Q = \sigma \left({T}^{4} - T {'}^{4}\right)$
$= \sigma \left({T}^{2} - T {'}^{2}\right) \left({T}^{2} - T {'}^{2}\right)$
$= \sigma \left(T - T '\right) \left(T + T '\right) \left({T}^{2} + T {'}^{2}\right)$
$= \sigma \left(T - T '\right) \left({T}^{3} + {T}^{2} T ' + T T {'}^{2} + T {'}^{3}\right)$

If the excess temperature $T - T '$ be small, then $T$ and $T '$ are nearly equal. So,
$Q = \sigma \left(T - T '\right) \cdot 4 T {'}^{3}$
$= \beta \left(T - T '\right)$

So, $Q \propto \left(T - T '\right)$ which is Newton's law of cooling.