Two sphere A and B have same radius, but the heat capacity of A is greater than that of B. The surfaces of both are painted black. They are heated to the same temperature and allowed to cool. Then

Let $C_A$ and $C_B$ be their heat capacities. Both the spheres have the same surface area. At any given temperature $\theta$ they lose heat at the same rate.

$-\frac{dT}{dt}=\frac{\sigma eA\left(T^4-T_0^4\right)}{ms}$

$\Rightarrow -\frac{dT}{dt}=\frac{\sigma eA\left(T^4-T_0^4\right)}{C}$

Both are painted black, so e is same. Both have same radius, so surface area is same. Both have same initial temperature and kept in same surrounding.

$\begin{array}{l}\Rightarrow C_A{\left(-\frac{d\theta }{dt}\right)}_A=C_B{\left(-\frac{d\theta }{dt}\right)}_B\\ \Rightarrow \frac{{\left(-\frac{d\theta }{dt}\right)}_A}{{\left(-\frac{d\theta }{dt}\right)}_B}=\frac{C_B}{C_A}=cons\tan t\end{array}$

Since, $C_A>C_B$

${\left(-\frac{d\theta }{dt}\right)}_B>{\left(-\frac{d\theta }{dt}\right)}_A$

B cools faster than A.

Rate of cooling of B is fast but their ratio is constant.