# Black body heat transfer

A solid spherical black body of density $\rho$ and specific heat capacity $S$ has a radius of $R$. If the sphere is initially heated to a temperature of $400 \text{ K}$ and suspended inside a chamber whose walls are at almost $0 \text{ K}$, then what is the time required for the temperature of the sphere to drop down to $200 \text{ K}?$

If this value can be represented as $t = \dfrac ab R \rho S,$ where $a$ and $b$ are coprime positive integers, evaluate the value of $a+b$.

Details and Assumptions:

• The chamber is an insulated isothermal enclosure and thermal equilibrium is maintained during the whole process.
• Explicitly assume that the Stefan-Boltzmann constant is $\sigma= 6 \times 10^{-8} \text{ Wm}^{-2} \text{ K}^{-4}$.
×