Question #43360

Two identical perfectly black spherical resistors, each of surface area $A$, are placed in a vacuum chamber whose walls are maintained at absolute zero ($0\text{ K}$). The electrical resistance of each sphere varies with its absolute temperature $T$ according to the relation $R(T) = R_0 \left(\frac{T}{T_0}\right)$, where $R_0$ is the resistance at a reference temperature $T_0$. Resistor 1 is connected to an ideal constant current source $I_0$, while Resistor 2 is connected to an ideal constant voltage source $V_0$. It is given that $V_0 = I_0 R_0$. Let the initial power dissipated at temperature $T_0$ be $P_0 = I_0^2 R_0 = \frac{V_0^2}{R_0}$. If $P_0 = \eta (A \sigma T_0^4)$, where $\sigma$ is the Stefan-Boltzmann constant and $\eta$ is a dimensionless positive constant, what is the ratio of the steady-state power dissipated by Resistor 1 to the steady-state power dissipated by Resistor 2, $\frac{P_1}{P_2}$?