We present exact closed-form expressions and complete asymptotic expansions for the electrostatic force between two charged conducting spheres of arbitrary sizes. Using asymptotic expansions of the force, we confirm that even like-charged spheres attract each other at sufficiently small separation unless their voltages/charges are the same as they would be at contact. We show that for sufficiently large size asymmetries, the repulsion between two spheres increases when they separate from contact if their voltages or their charges are held constant. Additionally, we show that in the constant voltage case, this like-voltage repulsion can be further increased and maximized through an optimal lowering of the voltage on the larger sphere at an optimal sphere separation.

我们为任意大小的两个带电导电球之间的静电力提供了精确的闭合形式表达式和完整的渐近展开。使用力的渐近展开，我们确认，即使带相同电荷的球体也以足够小的间距相互吸引，除非它们的电压/电荷与接触时的相同。我们表明，对于足够大的尺寸不对称性，如果两个球体的电压或电荷保持恒定，则当它们从接触中分离出来时，它们之间的排斥力会增加。此外，我们表明，在恒定电压情况下，可以通过在最佳球体间距下以最佳方式降低较大球体上的电压，来进一步提高和最大化这种类似电压的排斥力。