Newton's law of cooling represents a special case of

According to Stefan's law, the net rate of heat loss by radiation from a body at temperature $T$ in surroundings at temperature $T_0$ is given by $P = e \sigma A (T^4 - T_0^4)$.

Let $T = T_0 + \Delta T$, where the temperature difference $\Delta T$ is very small compared to $T_0$.

Substituting this into Stefan's law:

$P = e \sigma A [(T_0 + \Delta T)^4 - T_0^4]$

P = e \sigma A T_0^4 \left[ \left(1 +\dfrac{\Delta T}{T_0}\right)^4 - 1\right]$<br><br>Using the binomial expansion for small \dfrac{\Delta T}{T_0} and neglecting higher-order terms:<br><br>$P \approx e \sigma A T_0^4 \left[1 + 4\dfrac{\Delta T}{T_0} - 1\right]$<br><br>$P \approx 4 e \sigma A T_0^3 \Delta T$<br><br>Since$4 e \sigma A T_0^3$is a constant for a given body and surroundings, we get$P \propto \Delta T.

This is Newton's law of cooling, which states that the rate of heat loss is directly proportional to the temperature difference between the body and its surroundings. Thus, Newton's law of cooling is a special case of Stefan's law.