An analogous situation occurs in the case of the second configuration. For configuration 1, because the resultant dipole vectors from which the multipoles are composed are directed along z ̂, the quantity B z is identically equal to zero in the horizontal plane, though it is depicted for completeness. The unit of dipole strength p is equal to 1 μ A m. The origin of the coordinate system is located at (0,0,0) cm and the horizontal surface for which data are drawn is located at a height of x = 5 cm above the y − z plane. Moreover, the color map scaling is identical for each field component as well as field magnitude so that differences in amplitude between configurations can be more easily visualized. Color map scaling is symmetric for each plot in the sense that the minimum and maximum in each color map have the same absolute value. 1), and those for a 21 3 are in the third column (configuration 3, gray arrows in Fig. 1), those for a 21 2 are in the second column (configuration 2, black arrows in Fig. The field components and magnitude for a 21 1 are in the first column (configuration 1, white arrows in Fig. Magnetic fields of the electric multipole sources in a conducting medium surrounded by an insulating spherical shell are also presented and the relevance of this calculation to cardiographic and encephalographic experimentation is discussed.įigure 4(Color online) Two-dimensional plots of the magnetic-field components ( B x, B y, and B z) and magnitude ( | B | ) due to the three alternative configurations of the electrically silent quadrupole a 21 in Fig. In this context, information derived from magnetic recordings of electrically silent, magnetically active multipoles can supplement electrical recordings for the purpose of studying the physiology of the brain. Moreover, electrically silent, magnetically active moments of higher order can be useful when cancellation due to superposition of fields can occur, since this situation leads to a substantial reduction in the measurable amplitude of the signal. Our results have important practical applications in cases where electrically silent sources that generate measurable magnetic fields are of interest. This implies a mathematical relationship of complementarity between electroencephalography and magnetoencephalography, although the theoretical result in question does not apply to the nonspherical case. In the context of a spherical brain source model, it has been mathematically demonstrated that the part of the neuronal current generating the electric potential lives in the orthogonal complement of the part of the current generating the magnetic potential. We demonstrate how the process can be extended to even higher-order terms in an electrically silent series of magnetic multipoles. Although measurements of the electric potential are not sufficient to determine uniquely the characteristics of a quadrupolar source, the radial component of the magnetic field can supply the additional information needed to resolve these ambiguities and to determine uniquely the configuration of dipoles required to specify the electric quadrupoles. Expressions for the magnetic fields of electric dipole and quadrupole current sources immersed in an infinite conducting medium are derived, and it is shown that two different point dipole distributions that are electrically equivalent have different magnetic fields. In this paper, we clearly demonstrate that the electric potential and the magnetic field can contain different information about current sources in three-dimensional conducting media.
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