Question #42736

A cell of electromotive force $E$ and temperature-dependent internal resistance $r(T)$ is connected across a variable external resistor $R$. The internal resistance varies with the cell's temperature $T$ according to the relation $r(T) = \frac{r_0}{1 + \alpha(T - T_0)}$, where $r_0$ is the internal resistance at the constant ambient temperature $T_0$, and $\alpha$ is a positive constant. The cell loses heat to the surroundings at a rate $P_{loss} = k(T - T_0)$, where $k$ is the heat transfer coefficient. The external resistance $R$ is continuously adjusted so that the electrical power delivered to it is always maximum. If the cell reaches a steady-state temperature $T_{ss}$, what is the steady-state temperature difference $\Delta T = T_{ss} - T_0$? (Assume $4kr_0 > E^2 \alpha$)