Question

A certain planet of radius $R$ is composed of a uniform material that, through radioactive decay, generates a net power $P$ such that heat current is $\frac{\Delta Q}{\Delta t} = P\frac{r^3}{R^3}$, where $r$ is radial distance from centre. This results in a temperature differential between the inside and outside of the planet as heat is transferred from the interior to the surface. The rate of heat transfer inside is governed by conduction. It is found that thermal conductivity $k$ is constant for the planet. For the following assume that the planet is in a steady state; temperature might depend on position, but does not depend on time. (Assuming black body radiation and emissivity is 1). Find an expression for the temperature difference between the surfaces of the planet $(T_s)$ and the centre of the planet $(T_c)$. If your answer is $\Delta T = \frac{\sigma R^m T_s^n}{dk}$ fill value of $(m + n + i)$.

Answer 1

9

Answer 2

The rate of heat transfer by conduction through a spherical shell is given by Fourier's Law: $\frac{\Delta Q}{\Delta t} = -k A \frac{dT}{dr}$. Given heat current $I(r) = P\frac{r^3}{R^3}$, we have $P\frac{r^3}{R^3} = -k (4\pi r^2) \frac{dT}{dr}$. This simplifies to $\frac{dT}{dr} = -\frac{P}{4\pi k R^3} r$. Integrating from $r=0$ to $r=R$ gives $T_s - T_c = -\frac{PR}{8\pi k}$. Thus, $\Delta T = T_c - T_s = \frac{PR}{8\pi k}$. The total power radiated from the surface is $P = \sigma (4\pi R^2) T_s^4$. Substituting $P$, we get $\Delta T = \frac{(\sigma 4\pi R^2 T_s^4) R}{8\pi k} = \frac{\sigma R^3 T_s^4}{2k}$. Comparing with $\Delta T = \frac{\sigma R^m T_s^n}{dk}$, we have $m=3$, $n=4$, and $d=2$. Assuming $i$ is a typo for $d$, $m+n+d = 3+4+2=9$.