Answer to Question #86760 in Optics for Ogooluwa

Question

A spherical black body of diameter 20cm emits radiant energy of a particular temperature T, calculate the temperature if the rate of emission is 192w

Answer 1

Given: $d = 0.2~\text{m};~P = 192~\text{W}. \\$

Stefan-Boltzmann Law: $P = {\sigma}AT^4; \\$

where $\sigma \approx 5.670367*10^{−8}~\frac{\text{W}}{\text{m}^2\text{K}^4}~$ is the Stefan–Boltzmann constant.

Area of a sphere surface is expressed via diameter as $A = {\pi}d^2.\\$

Using this knowledge and numerical values, the temperature in absolute (Kelvin) scale is

$T = \sqrt[4]{\frac{P}{{\sigma}A}} = \sqrt[4]{\frac{P}{{\sigma}{\pi}d^2}} = \sqrt[4]{\frac{192~\text{W}}{5.670367*10^{−8}~\frac{\text{W}}{\text{m}^2\text{K}^4}~*~3.14~*~(0.2~\text{m})^2}} \approx 405.21~\text{K}.$