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A power source of constant power S is kept at centre of three perfectly conducting concentric spherical shells of radii R, 2R and 3R having equilibrium temperature $T_{1}, T_{2}, T_{3}$ respectively. Assuming the spherical shells to be perfect black bodies, choose the CORRECT option(s) (Assume surrounding temperature to be zero).

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$T_{3} = {(\frac{S}{72 {πR}^{2} σ})}^{1 / 4}$

$T_{2} = {(\frac{13}{4})}^{1 / 4} T_{3}$

$T_{1} = {(\frac{49}{4})}^{1 / 4} T_{3}$

$T_{1} = T_{2} = T_{3}$

answer is B, C.

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Detailed Solution

Radiant power, $P = e {AσT}^{4}$

Net power radiated by shell 1, $P_{1} = e A_{1} σ ({T_{1}}^{4} - {T_{2}}^{4})$

Net power radiated by shell 2, $P_{2} = e A_{2} σ ({T_{2}}^{4} - {T_{3}}^{4})$

Net power radiated by shell 3, $P_{3} = e A_{3} σ ({T_{3}}^{4})$

During steady state,

$S = P_{1} = P_{2} = P_{3}$

After solving we get,

$T_{2} = {(\frac{13}{4})}^{1 / 4} T_{3}$

$T_{1} = {(\frac{49}{4})}^{1 / 4} T_{3}$

$T_{3} = {(\frac{S}{36 {πR}^{2} σ})}^{1 / 4}$

$T_{1} > T_{2} > T_{3}$

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