Question

find steady state temperature of black body disc having 2cm and mass 7g at 45 degree C SURROUNDING temperature, when kept initially at 28 degree C .hence determine Stefan's constant

Answer 1

$By\ Boltzman's\ Law \\ \\ \frac{\partial Q _{out} }{\partial t} =( \ A \Pi^{2} )\sigma (T^{4}_{sum_{}}-T^{4})\\ Gain\ rate\\ \frac{\partial Q _{in} }{\partial t} = \frac{\partial }{\partial t }mc\Delta T =mc\frac{\partial T}{\partial t}\\ Equating\ both \\ \frac{\partial Q _{out} }{\partial t} = \frac{\partial Q _{in} }{\partial t}\\ \frac{\partial T}{\partial t} =\frac{A\pi ^{2}}{mc}\sigma T_{sum}^{4}-\frac{A\pi ^{2}}{mc}\sigma T^{4}\\ \frac{\partial T}{\partial t} + \frac{A\pi ^{2}}{mc}\sigma T^{4} =\frac{A\pi ^{2}}{mc}\sigma T_{sum}^{4}\\ T (t) = \frac{1}{\sqrt[3]{\frac{3A\pi ^{2}} {mc}\sigma t +(\frac{1}{T-T_{sum}})3}} +T_{sum}\\ In\ steady -state\\ t \to \infty\\ T=T_{sum}=45^{\circ}C\\ on \ 318K\ (Steady)$