Question #42156

A metallic sphere of mass $m$ is placed in a vacuum chamber maintained at absolute zero. It loses heat to the surroundings via a thermal link such that the rate of heat loss is $P_{loss} = \gamma T$, where $T$ is the absolute temperature of the sphere and $\gamma$ is a positive constant. Simultaneously, the sphere is subjected to a time-varying magnetic field that induces eddy currents. The power dissipated by these eddy currents is $P_{eddy} = \frac{\beta}{T}$, where $\beta$ is a positive constant (this inverse dependence arises because the electrical resistivity of the metal is proportional to its absolute temperature, $\rho \propto T$). The specific heat capacity of the metal varies with temperature as $s(T) = s_0 T$. If the sphere is initially at a temperature $T_0 = 2\sqrt{\frac{\beta}{\gamma}}$, what is the time $t$ required for it to cool down to $T_f = \sqrt{\frac{2\beta}{\gamma}}$?