A sphere and a cube, both of copper have equal volumes and are black… | MHT CET 2023 (14 May, Shift 1)

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Thermal Radiation: Black Body, Stefan & Wien

A sphere and a cube, both of copper have equal volumes and are black. They are allowed to cool at same temperature and in same atmosphere. The ratio of their rate of loss of heat will be

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Answer: (c)

Same material, same temperature and surroundings, so rate of heat loss $\propto$ surface area $A$ .
Equal volumes: $\frac{4}{3}\pi r^3 = a^3$ , so $a = \left(\frac{4\pi}{3}\right)^{1/3} r$ .
$\frac{A_s}{A_c} = \frac{4\pi r^2}{6a^2} = \frac{2\pi/3}{(4\pi/3)^{2/3}}$
Cubing: $\left(\frac{A_s}{A_c}\right)^3 = \frac{8\pi^3/27}{16\pi^2/9} = \frac{\pi}{6}$
So the answer is (c) $\left(\frac{\pi}{6}\right)^{1/3}$ .

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