Micromagnetics with eddy currents

In this paper, we study the modified Landau-Lifshitz-Gilbert (LLG) equation for of a conducting, magnetic body. The modified LLG equations include the magnetic field due to eddy currents in the total effective magnetic field. We derive an expression for the magnetic field due to eddy current losses and show that it is well defined. We then show that the work done by the eddy currents in opposing the change of magnetization is a Rayleigh type dissipation function, and derive the modified LLG equations using the calculus of variations. Finally, we show that the modified LLG equations lead to a decrease in the Gibbs energy. This implies that the LLG equations describes a dynamic process proceeding spontaneously forward in time.