Charge density on thin straight wire, revisited
Abstract
The question of the equilibrium linear charge density on a charged straight conducting "wire" of finite length as its crosssectional dimension becomes vanishingly small relative to the length is revisited in our didactic presentation. We first consider the wire as the limit of a prolate spheroidal conductor with semiminor axis a and semimajor axis c when a/c<<1. We then treat an azimuthally symmetric straight conductor of length 2c and variable radius r(z) whose scale is defined by a parameter a. A procedure is developed to find the linear charge density λ(z) as an expansion in powers of 1/Λ, where Λ≡ln(4c^{2}/a^{2}), beginning with a uniform line charge density λ_{0}. We show, for this rather general wire, that in the limit Λ>>1 the linear charge density becomes essentially uniform, but that the tiny nonuniformity (of order 1/Λ) is sufficient to produce a tangential electric field (of order Λ^{0}) that cancels the zerothorder field that naively seems to belie equilibrium. We specialize to a right circular cylinder and obtain the linear charge density explicitly, correct to order 1/Λ^{2} inclusive, and also the capacitance of a long isolated charged cylinder, a result anticipated in the published literature 37 years ago. The results for the cylinder are compared with published numerical computations. The secondorder correction to the charge density is calculated numerically for a sampling of other shapes to show that the details of the distribution for finite 1/Λ vary with the shape, even though density becomes constant in the limit Λ→∞. We give a second method of finding the charge distribution on the cylinder, one that approximates the charge density by a finite polynomial in z^{2} and requires the solution of a coupled set of linear algebraic equations. Perhaps the most striking general observation is that the approach to uniformity as a/c→0 is extremely slow.
 Publication:

American Journal of Physics
 Pub Date:
 September 2000
 DOI:
 10.1119/1.1302908
 Bibcode:
 2000AmJPh..68..789J
 Keywords:

 01.50.i;
 41.20.Cv;
 Educational aids;
 Electrostatics;
 Poisson and Laplace equations boundaryvalue problems