Question

How do you derive Newtons Law of Cooling?

Answer 1

One way to derive it would be from Stefan's law.

Consider an object (which we, for theoretical purposes assume to be a black body) at a temperature `T` and is surrounded by an environment of constant temperature `T_0`.

Then by Stefan's law, the heat emitted from the object to it's surroundings (in unit time) would be proportional to `(T^4 - T_0^4)`

Thus, `Q = A(T^4 - T_0^4)`

Now, `(T^4 - T_0^4) = (T^2 + T_0^2)(T^2 - T_0^2)`

`implies (T^4 - T_0^4) = (T^2 + T_0^2)(T + T_0)(T - T_0)`

But, `(T^2 + T_0^2)(T + T_0) = T^3 + T^2T_0 + T_0^2T + T_0^3)`

Now, for small differences between `T` and `T_0`, we get approximately,

`(T^2 + T_0^2)(T + T_0) = 4T_0^3` which is a constant since the surrounding is at constant temperature. (Let us denote it by `alpha`)

Using this result,

`Q = Aalpha(T - T_0)`

Thus, `Q = C(T - T_0)` (where `C = Aalpha`)

This proves Newton's law of cooling.