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Q.

A point source of light of power P₀ is placed at a distance of 4m from the centre of a thin hemispherical shell as shown in the figure, the shell has a radius of 3m and it behaves like a perfect black body. If the temperature of the hemisphere is related to the power of the source as $T^{4} = \frac{P_{0}}{n π σ}$ where is the Stefan’s constant, then find the value of $\frac{n}{10}$

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answer is 27.

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Total power emitted by shell due to radiation $P_{o u t} = σ (2 π R^{2} + π R^{2}) T^{4} = 3 π R^{2} σ T^{4}$

Power gained by the shell from $P_{0}$ : $P_{i n} = P_{0} \frac{2 π R^{2} (1 - \cos 37 °)}{4 π R^{2}} = \frac{P_{0}}{10}$
Equating: $\frac{P_{0}}{10} = 3 π R^{2} σ T^{4}$
Gives $T^{4} = \frac{P_{0}}{270 π σ}$
$n = \frac{270}{10} = 27$

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