Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2018 / Paper 1 / Section A(iii) / 6 / iii / Solution

At fixed temperature difference and constant internal resistance,Differentiation givesIt is positive below $R_L=R$ and negative above it, while power vanishes for an open or shorted load. Therefore the maximum power transfer theorem givesAt this point $I=V_{\mathrm{oc}}/(2R)$ and half the electromotive force drops across the load. Maximum electrical power is not generally maximum thermal-conversion efficiency. If changing current also changes the temperature difference because the thermal source has finite impedance, the fixed-$\Delta T$ optimization must be reconsidered.