An apparent paradox concerning the field of an ideal dipole

Abstract

The electric or magnetic field of an ideal dipole is known to have a Dirac delta function at the origin. The usual textbook derivation of this delta function is rather ad hoc and cannot be used to calculate the delta-function structure for higher multipole moments. Moreover, a naive application of Gauss's law to the ideal dipole field appears to give an incorrect expression for the dipole's effective charge density. We derive a general result for the delta-function structure at the origin of an arbitrary ideal multipole field without using any advanced techniques from distribution theory. We find that the divergence of a singular vector field can contain a derivative of a Dirac delta function even if the field itself does not contain a delta function. We also argue that a physical interpretation of the delta function in the dipole field previously given in the literature is perhaps misleading and may require clarification.

Generalized electromagnetism of subdimensional particles: A spin liquid story

“…We can then take ∂ 2 of this quantity by appealing to the ordinary Poisson equation and the charge distribution of an ordinary dipole. (Directly differentiating this potential is actually quite subtle, for distributional reasons, but a direct calculation yields the same results 21 .) The result is:…”

The Funk-Hecke formula, harmonic polynomials, and derivatives of radial distributions

“…Harmonic polynomials play a fundamental role in the ideas of the late professor Stora on convergent Feyman amplitudes, particularly in his work with Nikolov and Todorov [20]; this can be also seen in the recent article of Várilly and Gracia-Bondía [25]. Parker [21] has pointed out the correct formulas obtained for multipole potentials built from harmonic polynomials, while the author has shown that such multipole potentials have remarkable properties with respect to 2010 Mathematics Subject Classification: 46F10, 33C55. Submitted November 10, 2016.…”

“…Such derivatives, particularly in the case of regularizations of power potentials, are very important in Mathematical Physics [14,19,21], starting with the celebrated Frahm formulas [11] that have become standard material in textbooks [15]. The general derivatives of any order of regularizations of power potentials are available [17], so that, in principle one could evaluate (1.1) for any polynomial p for this type of radial distributions f, but the formulas simplify substantially precisely if p is harmonic.…”

The Gauss decomposition of products of spherical harmonics

“…Spherical harmonics have been studied in detail for centuries, as one can see in the texts [3,10,16], but we would like to point out the recent interest in harmonic polynomials in Mathematical Physics as they play a pivotal role in Stora's fine notion of divergent amplitudes [14,18], as well as in several aspects of the theory of multipoles [5,15]. The formulas given in this article are of general interest, but they will be particularly useful in these areas as well as in Fourier analysis, as needed in Mathematical Physics [1,2], and in integral geometry [9].…”