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A solid aluminium sphere and a solid copper sphere whose radius is twice that of the aluminium sphere are heated to the same temperature and then allowed to cool in identical surroundings. The emissivity of both spheres is the same. Given specific heat capacity of aluminium = 900 J/kg-degC, of copper = 390 J/kg-degC, and density of copper = 3.4 times the density of aluminium, find (a) the ratio of the rate of heat loss of the aluminium sphere to that of the copper sphere, and (b) the ratio of the rate of fall of temperature of the aluminium sphere to that of the copper sphere.

1. (a) 1/4 and (b) 2.95
2. (a) 1/2 and (b) 1.47
3. (a) 1/4 and (b) 0.34
4. (a) 1/8 and (b) 2.95

Correct answer: (a) 1/4 and (b) 2.95

Solution

By Stefan's law the rate of heat loss is proportional to the surface area when emissivity, body temperature and surroundings are identical. Surface area scales as r², so the larger copper sphere loses heat faster. The rate of fall of temperature is the heat-loss rate divided by the heat capacity (mass times specific heat), and mass scales as density times r³, so this ratio depends on density, radius and specific heat.

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