Symmetry of electrical conduction

The resistance of a conductor measured in a four-probe setup is invariant if the exchange of the voltage and current sources is accompanied by a magnetic field reversal. We present a derivation of this theorem. The reciprocity of the resistances is linked directly to the microscopic reciprocity of the S-matrix, which describes reflection at the sample and transmission through the sample. We demonstrate that this symmetry holds for a conductor with an arbitrary number of leads. Since leads act like inelastic scatterers, consideration of a many-probe conductor also implies that the reciprocity of resistances is valid in the presence of inelastic scattering. Various conductance formulae are discussed in the light of the reciprocity theorem. Finally, we discuss some implications of our results for the nature of a voltage measurement and point to the difference between chemical potentials and the local electric potential.