Generalization of Faraday’s Law to Include Nonconservative Spin Forces

Abstract

The usual Faraday's Law $\mathcal{E}=\ensuremath{-}d\ensuremath{\Phi}/dt$ determines an electromotive force $\mathcal{E}$ which accounts only for forces resulting from the charge of electrons. In ferromagnetic materials, in general, there exist nonconservative spin forces which also contribute to $\mathcal{E}$. These might be included in Faraday's Law if the magnetic flux $\ensuremath{\Phi}$ is replaced by $[\ensuremath{\hbar}/(\ensuremath{-}e)]\ensuremath{\gamma}$, where $\ensuremath{\gamma}$ is a Berry phase suitably averaged over the electron spin direction. These contributions to $\mathcal{E}$ represent the requirements of energy conservation in itinerant ferromagnets with time dependent order parameters.