The question of the equilibrium linear charge density on a charged straight
conducting "wire" of finite length as its cross-sectional dimension become
s vanishingly small relative to the length is revisited in our didactic pre
sentation. 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 variabl
e radius r(z) whose scale is defined by a parameter a. A procedure is devel
oped to find the linear charge density lambda(z) as an expansion in powers
of 1/Lambda, where Lambda = 1n(4c(2)/a(2)), beginning with a uniform line c
harge density lambda(0). We show, for this rather general wire, that in the
limit Lambda>>1 the linear charge density becomes essentially uniform, but
that the tiny nonuniformity (of order 1/Lambda) is sufficient to produce a
tangential electric field (of order Lambda(0)) that cancels the zeroth-ord
er 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/Lambda(2) inclusive, and also the capacitance of a long isolated
charged cylinder, a result anticipated in the published literature 37 year
s ago. The results for the cylinder are compared with published numerical c
omputations. The second-order correction to the charge density is calculate
d numerically for a sampling of other shapes to show that the details of th
e distribution for finite 1/Lambda vary with the shape, even though density
becomes constant in the limit Lambda-->infinity. We give a second method o
f finding the charge distribution on the cylinder, one that approximates th
e charge density by a finite polynomial in z(2) and requires the solution o
f a coupled set of linear algebraic equations. Perhaps the most striking ge
neral observation is that the approach to uniformity as a/c-->0 is extremel
y slow. (C) 2000 American Association of Physics Teachers.