Copper sphere is suspended in an evacuated chamber maintained at 300 K. The sphere is maintained at a constant temperature of 500 K by heating it electrically. A total of 210 W of electric power is needed to do it. When half of surface of the sphere is completely blackened, 630 W is needed to maintain the same temperature of the sphere. The emissivity of copper is

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$\frac{1}{5}$

$\frac{1}{2}$

$\frac{1}{3}$

$\frac{1}{4}$

answer is D.

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Detailed Solution

$210 = σ e A (500^{4} - 300^{4}) \dots \dots (i)$
$630 = σ e A (\frac{A}{2}) (500^{4} - 300^{4}) + σ (1) \frac{A}{2} (500^{4} - 300^{4}) \dots \dots (i i)$
$\frac{210}{630} = \frac{e}{\frac{1}{2} + \frac{e}{2}}$
$\frac{1}{3} = \frac{e}{\frac{1}{2} + \frac{e}{2}}$
$\frac{1}{6} + \frac{e}{6} = e$
$\Rightarrow \frac{1}{6} = e - \frac{e}{6} = \frac{5 e}{6}$
$e = \frac{1}{5}$