Electric field induced in a spherical volume conductor from arbitrary coils: application to magnetic stimulation and MEG.

A mathematical method is presented that allows fast and simple computation of the electric field and current density induced inside a homogeneous spherical volume conductor by current flowing in a coil. The total electric field inside the sphere is computed entirely from a set of line integrals performed along the coil current path. Coils of any closed shape are easily accommodated by the method. The technique can be applied to magnetic brain stimulation and to magnetoencephalography. For magnetic brain stimulation, the total electric field anywhere inside the head can be easily computed for any coil shape and placement. The reciprocity theorem may be applied so that the electric field represents the lead field of a magnetometer. The finite coil area and gradiometer loop spacing can be precisely accounted for without any surface integration by using this method. The theory shows that the steady-state, radially oriented induced electric field is zero everywhere inside the sphere for ramping coil current and highly attenuated for sinusoidal coil current. This allows the model to be extended to concentric spheres which have different electrical properties.