If the source of a field that satisfies Poisson’s equation can be written as the divergence of a vector s, then a scalar multipole expansion of the source can be evaluated in terms of s, which is a dipole density. A multipole expansion in terms of the dipole density can be computed about different origins. This allows us to evaluate the expansion of a dipole displaced from the origin and find a method of approximating some multipole expansions by displaced dipoles. In many physical applications it is known that the source is a displaced dipole, and we can find its location from a multipole expansion at some convenient location. It is possible to derive pictures in terms of dipole densities that in the proper limit become the individual multipoles. There are, however, ambiguities in that for some multipoles more than one picture gives the proper field.

Scalar multipole expansions have been widely used in many areas of physics as general solutions to Laplace’s equation. However, comparison of work by different authors is often complicated by the existence of several different forms of multipole expansions in different coordinate systems. In this paper we compare the spherical harmonic, Taylor’s series, direction cosine and traceless tensor expansions and present models of dipole configurations that in the proper limit give the individual multipoles. These pictures simplify the comparison of the series and aid in interpretation of the various multipole moments.

In this article, we clearly demonstrate that the electric potential and the magnetic field can contain different information about current sources in three-dimensional conducting media. 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. 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. We demonstrate how the process can be extended to even higher order terms in an electrically silent series of magnetic multipoles. 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. This implies a mathematical relationship of complementarity between electroencephalography (EEG) and magnetoencephalography (MEG), although the theoretical result in question does not apply to the non-spherical case (Dassios (2008), Math Med Biol, vol. 25, p. 133). Our results have important practical applications in cases where electrically silent sources that generate measurable magnetic fields are of interest. 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. 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. 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.

The influence of the interplanetary magnetic field (IMF) orientation on the transport of low‐energy ions injected from the ionosphere is investigated using three‐dimensional particle codes. It is shown that, unlike the auroral zone outflow, the ions originating from the polar cap region exhibit drastically different drift paths during southward and northward IMF. During southward IMF orientation, a “two‐cell” convection pattern prevails in the ionosphere, and three‐dimensional simulations of ion trajectories indicate a preferential trapping of the light ions (H+) in the central plasma sheet, due to the wide azimuthal dispersion of the heavy ions (O+). In contrast, for northward IMF orientation, the “four‐cell” potential distribution predicted in the ionosphere imposes a temporary ion drift toward higher L shells in the central polar cap. In this case, while the light ions can escape into the magnetotail, the heavy ions can remain trapped, featuring more intense acceleration (from a few electron volts up to the keV range) followed by precipitation at high invariant latitudes, as a consequence of their further travel into the tail.

“The Coach” is a web-based tool developed to guide students through the technical writing process. It provides instruction about form as well as critique of different aspects of the students’ writing. It goes beyond the Microsoft word spell check and grammar check. It gives feedback about writing complexity and appropriateness for different word choices in a technical document. It also gives background about the appropriate contents for technical writing in addition to example documents. The latter is extremely important for the novice writer who may not have much experience in working with technical reports. The initial document type in “The Coach” is a lab report. If the lab report can be developed into the web-based tool, other forms will be more easily implemented. In addition to developing the website, the development team is preparing a document and a video for a professor to use to instruct students on the use of “The Coach.” The instructional materials and “The Coach” were beta tested with a freshman engineering class. A baseline writing sample was collected before the introduction of “The Coach.” Students in some sections were instructed in use of “The Coach,” and other sections were controls. Additional beta testing is ongoing.