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The ratio of thermal capacities of two spheres A and B, if their diameters are in the ratio $1 : 2$ , densities in the ratio $2 : 1$ and specific heats in the ratio of $1 : 3$ , will be

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1 : 6

1 : 12

1 : 3

1 : 4

answer is B.

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Detailed Solution

H_{A} : H_{B} = ? d_{A} : d_{B} = 1 : 2

$ρ_{A} : ρ_{B} = 2 : 1, S_{A} : S_{B} = 1 : 3$

$r = \frac{d}{2}$

$H = ms = ρ vs = \frac{4}{3} π r^{3} s ρ$

$\frac{H_{A}}{H_{B}} = \frac{ρ_{A} S_{A} {(\frac{d_{A}}{2})}^{3}}{ρ_{B} S_{B} {(\frac{d_{B}}{2})}^{3}}$

$= \frac{2 \times 1 \times {(\frac{1}{2})}^{3}}{1 \times 3 \times {(\frac{2}{2})}^{3}} = \frac{2}{8} \times \frac{1}{3} \times 1 = \frac{1}{12}$

$∴ H_{A} : H_{B} = 1 : 12$

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