Charge density on thin straight wire, revisited

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 $a/c<<1.$ 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 $Λ≡ln(4c2/a2),$ beginning with a uniform line charge density $λ0.$ 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 $Λ0)$ 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 $1/Λ2$ 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 $z2$ 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.