We address the electrostatic problem of a thin, curved, and cylindrical conductor (a conducting filament) and show that the corresponding linear charge density slowly tends to uniformity as the inverse of the logarithm of a characteristic parameter, which is the ratio of the diameter to the smaller of the length and minimum radius of curvature of the filament. An alternative derivation of this result based on energy minimization is given. These results follow from a general asymptotic analysis of the electric field components and potential near a charged filament in the limit of vanishing diameter. It is found that the divergent parts of the radial and azimuthal electric field components are determined by the local charge density, while the axial component is determined by the local dipole density. For a straight filament our results reduce to those known for conducting needles. For curved filaments, the configuration of charges and fields is no longer azimuthally symmetric, and there is an additional length scale arising from the finite radius of curvature of the filament. The basic uniformity result survives the added complications, which include an azimuthal variation in the surface charge density of the filament. As with the variations in linear charge density along the filament, the azimuthal variations vanish with the characteristic parameter, only more rapidly. These findings allow us to derive an asymptotic formula for the capacitance of a curved filament that generalizes a result first obtained by Maxwell. The examples of a straight filament with uniform and linearly varying charge densities and a circular filament with a uniform charge distribution are treated analytically and found to agree with the general analysis. Numerical calculations illustrating the slow convergence of linear charge distribution to uniformity for an elliptical filament are presented. An interactive computer program implementing and animating the numerical calculations is available.

Topics
Electrostatics, Computer programming, Fourier analysis, Integral calculus, Asymptotic analysis, Coordinate system, Educational assessment, Mammals, Corrections, Public policy and governance
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Also see Jackson’s remarks on Maxwell’s pioneering work (Ref. 5).
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Note that this regularity condition (existence of a nonvanishing continuously differentiable tangent) is stronger than smoothness, which requires the existence of a nonvanishing continuous tangent.
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Strictly speaking, this statement is our definition of a curved circular cylinder.
9.
We will refer to a charged filament satisfying these requirements as “regular.” Needless to say, smoothness characteristics of λ(s) under equilibrium conditions are determined by those of R(s) and are not an independent issue.
10.
The device of using a line charge to describe the fields of the physical filament was also used by Andrews (Ref. 3) and Jackson (Ref. 4).
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Here the length parameter for the physical conductor is defined to be the same as that of the curve R(s) within.
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14.
Recall that 2η(α,s0,Φ0) is the diameter of the filament in units of L⁠, the smaller of the length and minimum radius of curvature of the filament, and that κ(s0)L is a pure number that never exceeds unity.
15.
This result is traditionally established by the argument that electrostatic equilibrium precludes the motion of charges within a conductor.
16.
If present, grounded conductors can be regarded as one with the Earth and treated as another insulated conductor for the purpose of this derivation.
17.
This is the conclusion of Thomson’s theorem. See, for example,
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18.
The vanishing of the first-order variations guarantees a stationary configuration, not necessarily a minimum. However, the minimum nature of the result here is clear on physical grounds and can be ascertained by examining the second-order variations.
19.
Calculations of the fields of straight filaments appear in various forms in the literature (Refs. 3 and 4).
20.
Specifically, we require that 1/2−|z/L|⪢ϵ⁠.
21.
Mathematically, the controlling parameter L⁠, which is the smaller of the length and minimum radius of curvature of the filament, would cease to exist, or perhaps vanish, at such points, in violation of our regularity conditions in Sec. II A. Physically, a “corner” essentially amounts to a point where the radius of curvature of the filament is smaller than, or comparable to, its diameter so that ϵ≳1⁠, far from the limit required for uniformity.
22.
The MATHEMATICA 7 program is posted at ⟨www.csus.edu/indiv/p/partovimh/pg.nb⟩.
© 2009 American Association of Physics Teachers.
2009
American Association of Physics Teachers
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