Charge density on thin straight wire, revisited
American Journal of Physics
The question of the equilibrium linear charge density on a charged straight conducting "wire" of finite length as its cross-sectional 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 semi-minor axis a and semi-major axis c when a/cϽϽ1. We then treat an azimuthally symmetric straight conductor of length 2c and variable radius r(z) whose scale is defined by a
... efined by a parameter a. A procedure is developed to find the linear charge density (z) as an expansion in powers of 1/⌳, where ⌳ϵln(4c 2 /a 2 ), beginning with a uniform line charge density 0 . We show, for this rather general wire, that in the limit ⌳ϾϾ1 the linear charge density becomes essentially uniform, but that the tiny nonuniformity ͑of order 1/⌳͒ is sufficient to produce a tangential electric field ͑of order ⌳ 0 ) that cancels the zeroth-order 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/⌳ 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 second-order 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/⌳ vary with the shape, even though density becomes constant in the limit ⌳→ϱ. 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.