The electrostatic force between two conducting spheres is solved exactly from Coulomb's law. The force is evaluated in two steps: (i) the charge distribution on each of the spheres is calculated as a function of separation distance by way of a solution to the Fredholm integral equation; (ii) the effect of the known charge distribution is integrated to obtain the electrostatic force. Evaluation of the electrostatic force is fast because the series expression for the charge distribution is `super converging', a characteristic trait for some Fredholm-type equations.
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Khachatourian et al. (1999) studied this question.
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