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 We then treat an azimuthally symmetric straight conductor of length and variable radius whose scale is defined by a parameter a. A procedure is developed to find the linear charge density as an expansion in powers of 1/Λ, where beginning with a uniform line charge density We show, for this rather general wire, that in the limit 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 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 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/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 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 is extremely slow.
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September 01 2000
Charge density on thin straight wire, revisited
J. D. Jackson; Charge density on thin straight wire, revisited. Am. J. Phys. 1 September 2000; 68 (9): 789–799. https://doi.org/10.1119/1.1302908
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