Distributions in spherical coordinates with applications to classical electrodynamics

Abstract

A general and rigorous method to deal with singularities at the origin of a polar coordinate system is presented. Its power derives from a clear distinction between the radial distance and the radial coordinate variable, which makes that all delta-functions and their derivatives are automatically generated, and insures that the Gauss theorem is correct for any distribution with a finite number of isolated point-like singularities.The method is applied to the Coulomb field, and to show the intrinsic differences between the dipole and dimonopole fields in classical electrodynamics.In all cases the method directly leads to the general expressions required by the internal consistency of classical electrodynamics.

Products of Harmonic Polynomials and Delta Functions

“…Products of this kind are found very frequently in many areas, such as the distributional solution of differential equations [11], [15], [16], and in many of the applications of distribution theory, particularly in Mathematical Physics 1 [2], [3], [10], [12], [13], [17], [18]. For example, the known formulas for the derivatives of power potentials [5], [6] often need to be multiplied by polynomials in the computations performed in the Physics literature [10], [14].…”

The classical point electron in Colombeau’s theory of nonlinear generalized functions

“…There is however comparatively little use of them in electrodynamics [16,17,18,19,20], despite that products of distributions arise naturally in that domain, especially when attempting to calculate the self-interaction terms arising in any problem dealing with point-like charged particles, e.g., the self-force and the selfenergy, which are well-known to be divergent quantities. In fact, in the few cases in which generalized functions other than Schwartz distributions have been used, the objective has been to provide a better mathematical justification for already known results [21,22].…”

“…519], [18], E( r ) = e r r 3 Υ(r) − e r r 2 Υ ′ (r), (1.1) where Υ is a G-function (defined in Sec. 5) with properties similar to Heaviside's step function, so that the G-function Υ ′ has properties similar to Dirac's δ-function.…”

“…Applying the same principles to the magnetic dipolar field, which is wellknown to very precisely characterize the intrinsic magnetic dipole moment of elementary particles, one finds [18] H( r ) = 3 r( µ · r) − µ r 5 Υ(r) + r × ( µ × r) r 4 Υ ′ (r), (1.2) where in contrast to that in (1.1), the additional δ-like contribution is well known.…”

On the electromagnetic momentum of static charge and steady current distributions

“…For example, the complete field strength of a point magnetic dipole of magnetic moment m is [3] B d ( r) = 3 u( m · u) r 3 − m r 3 Υ(r) + u × ( m × u) r 2 δ(r), (1.4) where u is the unit vector in the direction of r, and, as will be recalled in section 3, the generalised function Υ insures that differentiation at the position of the singularities properly leads to the δ functions which arise when calculating the fields and currents according to Maxwell's equations. For instance, after volume integration, the δ singularity in (1.4) gives the δ-like term discussed by Jackson in [10, p.184], which is essential in calculating the hyperfine splitting of atomic states [7].…”