A black coloured solid sphere of radius R and mass M is inside a cavity with a vacuum inside. The walls of the cavity are maintained at temperature…

$$\frac{d\theta }{dt}=-\frac{\sigma Ae}{Ms}({\theta }^{4}-{\theta }_{0}^{4})$$dtdθ​=−MsσAe​(θ4−θ04​)

$$\frac{d\theta }{dt}=-\frac{\sigma Ae}{M\alpha {\theta }^{3}}({\theta }^{4}-{\theta }_{0}^{4})$$dtdθ​=−Mαθ3σAe​(θ4−θ04​)

$$\int \frac{{\theta }^{3}}{({\theta }^{4}-{\theta }_{0}^{4})}d\theta =-\frac{\sigma Ae}{M\alpha }\int dt$$∫(θ4−θ04​)θ3​dθ=−MασAe​∫dt

Let $({\theta }^{4}-{\theta }_{0}^{4})=x$(θ4−θ04​)=x

$$4{\theta }^{3}d\theta =dx$$4θ3dθ=dx

$$\int \frac{dx}{4x}=-\frac{\sigma Ae}{M\alpha }{[t]}_{0}^{t}$$∫4xdx​=−MασAe​[t]0t​

$$\frac{1}{4}[\mathrm{ln}x]=\frac{M\sigma Ae}{M\alpha }{[t]}_{0}^{t}$$41​[lnx]=MαMσAe​[t]0t​

$$\frac{1}{4}[\mathrm{ln}({\theta }^{4}-{\theta }_{0}^{4})]\begin{matrix}2{T}_{0} \\ 3{T}_{0}\end{matrix}=-\frac{\sigma Ae}{M\alpha }t$$41​[ln(θ4−θ04​)]2T0​3T0​​=−MασAe​t

$$\frac{1}{4}\mathrm{ln}(\frac{{(2{T}_{0})}^{4}-{T}_{0}^{4}}{{(3{T}_{0})}^{4}-{T}_{0}^{4}})=-\frac{\sigma Ae}{M\alpha }t$$41​ln((3T0​)4−T04​(2T0​)4−T04​​)=−MασAe​t

$$t=-\frac{\alpha M}{4\sigma Ae}\mathrm{ln}(\frac{15{T}_{0}^{4}}{80{T}_{0}^{4}})$$t=−4σAeαM​ln(80T04​15T04​​)

$$=-\frac{\alpha M}{4\sigma Ae}\mathrm{ln}(\frac{3}{16})$$=−4σAeαM​ln(163​)

$=-\frac{\alpha M}{4\sigma (4\pi {R}^{2})e}\mathrm{ln}(\frac{3}{16})$=−4σ(4πR2)eαM​ln(163​) where $e=1$e=1

$$t=\frac{\alpha M}{16\pi \sigma {R}^{2}}\mathrm{ln}(\frac{16}{3})$$t=16πσR2αM​ln(316​)