Question

a hemispherical shell has temp of base area 30k find the equilibrium temperature of curved surface surrounding temp is 0

Answer 1

The equilibrium temperature of the curved surface is $\frac{30}{2^{1/4}} K$.

Answer 2

At equilibrium, heat absorbed by the curved surface equals heat radiated. Heat is transferred from the base (at $T_{base}$) to the curved surface (at $T_{curved}$) through the shell. Heat is radiated from the curved surface to the surroundings (at $T_{surr}=0K$). Assuming the rate of heat transfer from the base is proportional to its area $A_{base}$ and $T_{base}^4$, and the rate of heat radiated is proportional to the curved area $A_{curved}$ and $T_{curved}^4$, we set these rates equal. $Q_{in} \propto A_{base} T_{base}^4$ and $Q_{out} \propto A_{curved} T_{curved}^4$. Equating them ($A_{base} T_{base}^4 = A_{curved} T_{curved}^4$) and using $A_{curved} = 2 A_{base}$, we get $A_{base} T_{base}^4 = 2 A_{base} T_{curved}^4$, which simplifies to $T_{base}^4 = 2 T_{curved}^4$. Solving for $T_{curved}$, we find $T_{curved} = T_{base} / 2^{1/4}$. Substituting $T_{base} = 30K$, the equilibrium temperature of the curved surface is $30 / 2^{1/4} K$.