Analytic solutions for some basic problems in electricity involving intersecting sphere and circular cylinder pairs
Analytic solutions for some basic problems in electricity involving intersecting sphere and circular cylinder pairs
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电学中相交球体与圆柱体对的一些基本问题的解析解
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发表时间：
2012
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通讯作者：
D. Palaniappan
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作者：
D. Palaniappan
利用经典静电学中的电学反演方法，得到了球/柱重叠静电基本问题的精确解。本文考虑的问题包括：（1）一对相交于顶角为π/n（n为整数）的重叠导电球，置于一个常势场中;（2）一对相交于顶角为π/n（n为整数）的无限长导电圆柱，置于一个均匀场中;由两个正交相交的无限长圆柱体组成的复合混合几何体，其边界是导电表面和电介质表面的组合（具有混合边界条件）浸没在均匀场中。重复应用开尔文反演的基本思想，在每种情况下导出电势的解。一个精确的表达式中的两个半径，中心到中心的距离，和顶角的电容被发现的孪生球的几何形状。然后，容量被用来找到一个完美的吸收体放置在一个导热的环境中的较低的温度的稳态率系数。利用二维问题的精确解绘制了等势图，并讨论了等势图的特点。这里说明的简单方法可以作为教学工具和进一步计算的构建块。
The method of electrical inversion in classical electrostatics is employed to obtain exact solutions for basic electrostatic problems pertaining to overlapping spheres/cylinders. The problems considered here include (1) a pair of overlapping conducting spheres, intersecting at a vertex angle π/n, n an integer, placed in a constant potential field; (2) a pair of infinitely long conducting circular cylinders, intersecting at a vertex angle π/n, n an integer, placed in a uniform field; and (3) a composite hybrid geometry consisting of two orthogonally intersecting infinitely long circular cylinders whose boundary is a combination of conducting and dielectric surfaces (with mixed boundary conditions) submerged in a uniform field. Applying the basic idea of Kelvin’s inversion repeatedly, solutions for the electric potentials are derived in each case. An exact expression for the capacitance in terms of the two radii, center-to-center distance, and the vertex angle is found for the twin sphere geometry. The capacity is then used to find the steady-state rate coefficient of a perfectly absorbing body placed in a thermally conducting environment of lower temperature. The equipotentials are plotted using the exact solutions of the two-dimensional problems and their features are discussed as well. The simple method illustrated here can be useful both as a teaching tool and as a building block for further computations.