Question

A spherical tungsten piece of radius $1\text{cm}$ is suspended in an evacuated chamber maintained at $300\text{K}$. The piece is maintained at $1000\text{K}$ by heating it electrically. Find the rate at which the electrical energy must be supplied. The emissivity of tungsten is $0.30$ and the stefan constant, $\sigma =6.0\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}$

• A. $18\text{W}$
• B. $22\text{W}$
• C. $26\text{W}$
• D. $30\text{W}$

Answer 1

The correct option is B $22\text{W}$

Given:
Radius of the Tungsten sphere, $r={10}^{-2}\text{m}$
emissivity of Tungsten, $e=0.3$
Stephen's constant, $\sigma =6.0\times {10}^{-8}{\text{Wm}}^{-2}{\text{K}}^{-4}$

Surrounding Temperature, $T_1=300K$

Temperature of the tungsten sphere, $T_2=1000K$

Now,
Surface area of Tungsten sphere,$A=4\pi r^2$
$\Rightarrow A=4\pi ({10}^{-2}{)}^2$= $4\pi \times {10}^{-4}{\text{m}}^2$

Rate at which energy is emitted from the tungsten sphere,
${\left(\frac{\Delta Q}{\Delta t}\right)}_e=e\sigma A{T}_2^4$

Rate at which energy is absorbed by the tungsten from the surrounding, ${\left(\frac{\Delta Q}{\Delta t}\right)}_a=e\sigma A{T}_1^4$

So, the rate at which electrical energy must be supplied to maintain it at constant temperature is equal to the net rate of heat loss from the sphere.

i.e. electrical power supplied
$={\left(\frac{\Delta Q}{\Delta t}\right)}_e-{\left(\frac{\Delta Q}{\Delta t}\right)}_a$

$P=e\sigma A(T_2^4-T_1^4)$

$P=0.3\times 6\times {10}^{-8}\times 4\pi \times {10}^{-4}({\left(1000\right)}^4-{\left(300\right)}^4)$

$\therefore P=22.4\text{W}\approx 22\text{W}$

Hence, $\left(B\right)$ is the correct answer.