Preprint December 2020 1 A modification of Plonsey´s formula for biomagnetism Version with corrections Roberto Suárez-Antola Abstract: A new formula is obtained here to relate the emf induced in a coil of a measuring system, with two fields: the so called impressed electric current density field (due to the action potentials in the membranes of the excitable cells located in the volume conductor of the biological tissues), and a certain stationary field, known as the magnetic lead field. The proposed formula modifies the one due to Robert Plonsey, published in 1972. After carrying out a review of the deduction of the original formula, its modification is deduced in the framework of electromagnetic theory and by analytical methods. The resultant modification emphasizes the weight assigned to the high frequencies in the induced emf. The deduction of the magnetic lead field is made here from a different approach. Some consequences of the modified formula are discussed and suggestions for further work are given. In Appendix A, the phasor field solutions of Maxwell equations corresponding to monochromatic oscillations in a general linear continuous medium are reviewed. Then, a general version of Lorentz reciprocity theorem for linear, nonhomogeneous, anisotropic, and lossy dispersive media is presented as a basis for deducing the modified formula proposed in this work. In Appendix B the restrictions that allow to apply the quasi-stationary approximation to electromagnetic theory in both, the quasi-electro-stationary and the magneto-quasi-stationary approaches are considered. Key words: Plonsey´s formula of biomagnetism, reciprocity theorem, magnetic field leads, impressed electric currents, inhomogeneous and anisotropic volume conductor, quasi- stationary approximations Introduction Organisms produce magnetic fields related with physiological electrical currents (such as currents associated with muscle contraction and the transfer of electrical signals through the nervous system) or with the magnetization of some body tissues. Compared to the Earth's magnetic field, they are quite weak, but can be measured with instruments sensitive enough and under appropriate conditions to eliminate disturbances of external origin (Hobbie and Roth, 2007). In the sixties of the last century the magnetic fields produced by the electrical activities of the heart and brain were measured. The magnetic fields were measured from the recording of the electromotive force induced by the fields, variable in time, in coils with millions of turns, locating the organism and the measurement system in isolated rooms (from the magnetic point of view) and averaging the signals (Baule and Mc Fee, 1965). Starting in the following decade, much more sensitive instruments, based on superconducting quantum interference (SQUID), began to be developed and used. Combined with the possibilities offered by multichannel electronics, new signal processing algorithms and mathematical models of the current sources, SQUIDs made it possible to develop basic research and clinical diagnostic methods (Cheyne, 2006). These methods are based on inferences about the distribution of electrical currents in biological tissues that can be obtained from measurements and interpretations of the magnetic fields produced by these currents.