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A solid copper sphere (density p and specific heat c) of radius .r at an initial temperature 200 K is suspended inside a chamber whose walls are at almost O K. The time required for the temperature of the sphere to drop to 100 K is

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a

7rρc72σ×10−6sec

b

7rρc272σ×10−6sec

c

r2ρc72σ×10−6sec

d

rρ2c2σ2×10−6sec

answer is A.

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Detailed Solution

mc−dTdt=σT4A ∴ dt=−mcdTσAT4Integrating, we gett=−mcσA∫200100 T−4dT ∴ t=43πr3ρcσAT−3(−3)200100 =4πr3ρc9σ4πr21(100)3−1(200)3=rρc9σ1(100)3−1(200)3=7rρc72σ×10−6sec

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A solid copper sphere (density p and specific heat c) of radius .r at an initial temperature 200 K is suspended inside a chamber whose walls are at almost O K. The time required for the temperature of the sphere to drop to 100 K is