ABSTRACT
A line charge in space corresponds to the generalized function two-dimensional unit impulse, and the corresponding in›uence or Green function for the Poisson equation speci”es the potential due to the line charge. ¡is method is distinct from, and leads to the same results as, the theory of the complex potential (Chapters I.12, 24, and 26), not only for monopoles, but also for dipoles, quadrupoles, and multiple expansions. Whereas the method using complex functions is restricted to plane problems, the use of generalized functions is valid regardless of spatial dimension; for example, the latter method extends (Section 9.2) readily from two (Chapter I.11) to three (Chapter 8) or higher (Sections 9.7 through 9.9) dimensions. ¡e three-dimensional unit impulse represents a point charge, and the associated in›uence or Green function speci”es the corresponding potential; the convolution integral expressing the principle of superposition then speci”es the potential due to an arbitrary distribution of charges. ¡is leads to the three-dimensional laws for the Coulombian force ”eld which are related to plane laws (Section I.24.4), but distinct (Section 8.3) from them. In space (Section 8.1), as in the plane (Section I.12.7), a dipole is the limit of two opposite monopoles whose strength increases inversely with distance; whereas the monopole is speci”ed by a unit impulse and is radially symmetric, a dipole involves the gradient of a unit impulse projected in the direction of the vector dipole moment and, thus, is axisymmetric (Section 8.2). A quadrupole is (Section 8.3) the limit of opposite dipoles and involves second-order derivatives of the unit impulse; since it has two axes, it is no longer generally axisymmetric, and the quadrupole moment is represented by a matrix. A three-dimensional charge distribution, like its two-dimensional counterpart, can be decomposed into a superposition of multipolar ”elds, with an important dižerence: (1) the two-dimensional multipoles have moments that are complex numbers; (2) the three-dimensional multipoles have moments of increasing complexity, namely, a scalar, a vector, a matrix, and a tensor, respectively, for mono-, di-, quadru-, and multipoles. ¡e dižerences between the electrostatic (magnetostatic) ”eld in space are greater than in the plane [Chapter I.24 (Chapter I.26)]: (1) the electrostatic ”eld is irrotational, and is always represented by a scalar potential in two (Section I.24.2), three (Sections 8.1 through 8.3), or higher (Sections 9.7 through 9.9) dimensions; (2) the magnetostatic ”eld is solenoidal and is represented by a pseudoscalar ”eld function in two dimensions (Section I.26.2) and by a vector potential (Section 8.4) in three dimensions; and (3) the three-dimensional vector potential reduces to a scalar stream function not only in the plane case but also in the three-dimensional axisymmetric case (Sections 6.2 through 6.7). Both the electrostatic (magnetostatic) ”elds have a multipolar expansion for the scalar (vector) potential [Section 8.3 (Section 8.4)]. ¡e lowest-order term in the vector multipolar expansion for the magnetostatic potential corresponds to a point current; it is speci”ed by a unit impulse like the electrostatic monopole, but has a vector direction
and is axisymmetric, corresponding to a magnetic dipole (Section 8.5). ¡e limit of two opposite magnetic dipoles is a magnetic quadrupole (Section 8.6) that (1) involves derivatives of the unit impulse as an electrostatic dipole; and (2) has a quadrupole moment speci”ed by a matrix, and its two axes imply it is generally not axisymmetric. ¡e method of images applies to electro(magneto-) static ”elds and extends from the plane to space, namely, there are identical (opposite) images on an insulating (conducting) boundary, for example, (1) for a point electric charge or current near a plane (Section 8.7), that has one image; (2) for a point charge between orthogonal (parallel) planes, that has three (or an in”nite number of) images (Section 8.8); and (3) for a point charge near a sphere (Section 8.9), using the axisymmetric sphere theorem (Section 6.7) involving the reciprocal point. ¡e point charge near an insulating boundary is analogous from the point of view of images (Sections 8.6 through 8.9) to a ›ow source/sink near a rigid wall and leads to continuous image distributions. ¡e continuous source distributions can be used to obtain the potential ›ow past bodies of arbitrary shape (Section 8.9).
