A Model for Multiple Metal Spheres in Oscillating Magnetic Fields Using Displaced Dipoles

In this article, we derive a magnetic dipole model for two identical, electrically conducting, and permeable macroscopic spheres that are exposed to an oscillating homogeneous magnetic field. Our model predicts both amplitude and phase of the induced field outside the spheres. The description is provided for parallel and transverse excitation relative to the axis through the sphere centers. This geometric decomposition allows the application of arbitrary excitation field directions. Our approach is based on one dipole per sphere. The origins of these secondary dipole fields are proposed to be found at positions slightly displaced from the sphere centers to consider the mutual interaction. This displacement and the resulting phase of the dipole moments strongly depend on the distance between the spheres as well as on complex-valued first- and second-order response factors, which contain material properties and the oscillation frequency. We demonstrate the usefulness of our displaced dipole model in terms of efficiency and accuracy compared to other computationally simple approaches.