Question

A solid copper sphere ( density $\rho$ and specific heat C) of radius r at an initial temperature 200K is suspended inside a chamber whose walls are at almost 0K. What is the time required for temperature of the sphere to drop to 100K?

Answer 1

The heat loss due to radiation will cause temperature to fall. Hence,

$e\sigma A{T}^4=\frac{dQ}{dt}$

Since $dQ=mCdT$

$e\sigma \left(4\pi r^2\right)T^4=\frac{\rho \left(\frac{4}{3}\pi r^3\right)CdT}{dt}$

$\frac{3e\sigma }{\rho rC}t={\int }_{200}^{100}\frac{dT}{T^4}$

$t=\frac{\rho rC}{9e\sigma }(1-\frac{1}{8}){10}^{-6}=\frac{7\rho rC}{72e\sigma }{10}^{-6}$