Charge density on thin straight wire, revisited
Charge density on thin straight wire, revisited
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DOI：
10.1119/1.1302908
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发表时间：
2000-09-01
影响因子：
0.9
通讯作者：
Jackson, JD
中科院分区：
文献类型：
--
作者：
Jackson, JD
在我们的教学演示中，重新讨论了有限长度的带电直导线上的平衡线电荷密度问题，因为它的横截面尺寸相对于长度变得非常小。我们首先考虑线作为具有半短轴a和半长轴c的长球形导体的极限，当a/c1时，线电荷密度变得基本均匀，但是微小的不均匀性（1/λ阶）足以产生切向电场（λ（0）阶），该切向电场抵消了零阶场，零阶场天真地似乎不符合平衡。我们专注于一个正确的圆柱体，并获得明确的线性电荷密度，正确的顺序1/λ（2）的包容性，以及一个长的孤立的充电圆柱体的电容，在37年前发表的文献中预期的结果。圆柱体的结果与发表的数值计算进行了比较。对电荷密度的二阶校正是针对其他形状的采样进行数值计算的，以显示有限1/Lambda的分布的细节随形状而变化，即使密度在极限Lambda->无穷大中变为常数。我们给出了第二种方法找到的电荷分布的圆柱体上，一个近似的电荷密度由一个有限的多项式在z（2），并需要一组耦合的线性代数方程组的解决方案。也许最引人注目的一般性观察是，达到a/c-->0的均匀性的方法是极其缓慢的。(C)2000年美国物理教师协会。
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/c1 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-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/Lambda(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/Lambda vary with the shape, even though density becomes constant in the limit Lambda-->infinity. 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. (C) 2000 American Association of Physics Teachers.