Question #42749

A potentiometer wire of length $L$, cross-sectional area $A$, and thermal conductivity $K$ is connected to a constant current source $I_0$. The ends of the wire at $x=0$ and $x=L$ are clamped to massive blocks maintained at a constant temperature $T_0$. The wire is kept in a vacuum, so heat is only lost via conduction to the ends. The electrical resistivity of the wire varies with temperature $T$ as $\rho(T) = \rho_0 [1 + \alpha (T - T_0)]$, where $\rho_0$ is the resistivity at $T_0$ and $\alpha$ is the temperature coefficient of resistivity. Assume the heat generated per unit volume is uniform and approximately equal to $\frac{I_0^2 \rho_0}{A^2}$. Two cells of EMFs $E_1$ and $E_2$ are balanced on this potentiometer wire at null points $x_1 = L/3$ and $x_2 = 2L/3$ respectively, measured from the $x=0$ end. Given the dimensionless parameter $\gamma = \frac{\alpha I_0^2 \rho_0 L^2}{2 K A^2}$, what is the ratio $\frac{E_1}{E_2}$?