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A solid sphere and a hollow sphere of the same material and size are heated to the same temperature and allowed to cool in the same surroundings. If the temperature difference between each sphere and its surroundings is $T$, then
The hollow sphere will cool at a faster rate for all values of $T$
The solid sphere will cool at a faster rate for all values of $T$
Both spheres will cool at the same rate for all values of $T$
Both spheres will cool at the same rate only for small values of $T$
Solution
(a) Rate of cooling $\frac{{\Delta \theta }}{t} = \frac{{A\varepsilon \sigma ({T^4} – T_0^4)}}{{mc}}$
As surface area, material and temperature difference are same, so rate of loss of heat is same in both the spheres. Now in this case rate of cooling depends on mass.
==> Rate of cooling $\frac{{\Delta \theta }}{t} \propto \frac{1}{m}$
( ${m_{solid}} > {m_{hollow}}$. Hence hollow sphere will cool fast.
