Question

Newton's law of cooling is a special case of

(a) Wien's displacement law
(b) Kirchhoff's law
(c) Stefan's law
(d) Planck's law

Answer 1

(c) Stefan's 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,
$u=\sigma A{T}^4$
Where $\sigma$ 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
$u=e\sigma A{T}^4$
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)
$u_0=e\sigma A{T_0}^4$

Thus, the net loss of thermal energy per unit time is
$$\begin{gathered} ∆u=u-u_0 \\ ∆u=e\sigma A(T^4-{T_0}^4)...\left(\mathrm{i}\right) \end{gathered}$$

Newton law of cooling is given by
$\frac{\mathit{d}T}{\mathit{d}t}=-bA(T-T_0)$

This can be obtained from equation (i) by considering the temperature difference to be small and doing the binomial expansion.