Charged line segments and ellipsoidal equipotentials

Abstract

This is a survey of the electrostatic potentials produced by charged straight-line segments, in various numbers of spatial dimensions, with comparisons between uniformly charged segments and those having non-uniform linear charge distributions that give rise to ellipsoidal equipotentials surrounding the segments. A uniform linear distribution of charge is compatible with ellipsoidal equipotentials only for three dimensions. In higher dimensions, the linear charge density giving rise to ellipsoidal equipotentials is counter-intuitive -the charge distribution has a maximum at the center of the segment and vanishes at the ends of the segment. Only in two dimensions is the continuous charge distribution intuitive -for that one case of ellipsoidal equipotentials, the charge is peaked at the ends of the segment and minimized at the center.

Definition and properties of logopoles of all degrees and orders

“…This form is similar to that used in Ref. [10] to describe uniformly charged line segments in 2m + 3 dimensions where the electrostatic force decays as 1/r 2m+1 . This aligns with the interpretation of L m −m as a distribution of transverse 2 m multipoles on the line segment which share this 1/r 2m+1 decay factor.…”

Screened Coulomb interactions of general macroions with nonzero particle volume

“…To show the generality of the framework, I consider the interaction potentials of two triaxial ellipsoids, for which a mapping to a spherical shell is needed, rather than a line charge. I hypothesize, however, that some metal spheroidal particles can be mapped to straight lines, because the (unscreened) isopotential surfaces of straight lines are prolate spheroids with specific aspect ratios [70]. It is straightforward to generalize the approach to mappings to ion-penetrable charged surfaces S with surface charge distribution σ s , and S parametrized by :…”

Charge densities for conducting ellipsoids

“…Moreover, for any conducting ellipsoid in three spatial dimensions, the charge per length is always constant when projected along any of the three principal axes, a feature that is perhaps the one elementary property that is easiest to keep in mind. However, in any other number of spatial dimensions, this last statement must be modified, as discussed recently in [9].…”

“…Finally, in an appendix, we discuss the geometry of hyperellipsoids from both intrinsic and extrinsic points of view. Overall, we have tried to present our discussion at a level suitable for use in graduate courses on electrostatics, as a supplement to more traditional material, similar to the presentations in two previous articles in this journal [9,10].…”