Question

A copper sphere is suspended in an evacuated chamber maintained at $300\text{K}$. The sphere is maintained at a constant temperature of $500\text{K}$ by heating it electrically. A total of $300\text{W}$ of electric power is needed to do it. When half of the surface of the copper sphere is completely blackened, $600\text{W}$ is needed to maintain the same temperature of the sphere. Calculate the emissivity(e) of copper.

Answer 1

Case (a) : Full copper sphere is maintained at $500K$


$P_{in}=P_{out}$
$300=\sigma eA(T_b^4-T_s^4)$ (using Stefan's law)

Here, $T_b=500K$ (temperature of body)
$T_s=300K$ (temperature of surroundings)

$300=\sigma eA({500}^4-{300}^4)$ -- (1)

Case (b) : Half copper + half body


$600=\sigma e\frac{A}{2}({500}^4-{300}^4)+\sigma \times 1\times \frac{A}{2}({500}^4-{300}^4)$ --- (2)

(emissivity $e=1$ for a black body)

(1) / (2) $\Rightarrow$

$\frac{300}{600}=\frac{\sigma eA({500}^4-{300}^4)}{\sigma e\frac{A}{2}({500}^4-{300}^4)+\sigma \times 1\times \frac{A}{2}({500}^4-{300}^4)}$

$\Rightarrow \frac{1}{2}=\frac{e}{\frac{e}{2}+\frac{1}{2}}$

$\Rightarrow e=\frac{1}{3}=0.33$