Electrostatic Force between Charged Conducting Spheres

This Demonstration calculates the electrostatic force between two conducting spheres held at constant voltage or constant charge. All the charge polarization effects are taken into account, even at small separations [1, 2]. Also output are the charges and voltages on the spheres, their overall electrostatic energy and the average electric field in the gap between them. The diagram is scaled so that even small spheres are visible. The distance ratio can be input as .

SNAPSHOTS

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DETAILS

Variables Used
is the distance ratio between spheres, where is their center-to-center distance and and are their radii. and are the voltages on the spheres. Without loss of generality, we choose to be positive and the larger of the two voltages. The two spheres can be interchanged or their polarities reversed without changing the force. and are the respective charges on the spheres. is the average electric field in the gap between the spheres. In experiments at sea level, sparking between spheres will occur around . Here is the force in newtons, proportional to , the permittivity of the surrounding medium. The permittivity of vacuum is used in the calculations.
Extreme Situations
When the spheres are in contact, set . Note that is automatically set equal to or in this case. For distances or radii outside of the ranges given, the input fields can be used for calculation. The output values will still be correct even though the slider bars will indicate that the value is out of range. Capacitance coefficients (in farads) can be calculated by choosing appropriate voltages on the spheres. For example, if and , then and .
Interesting Features
Two like-charged spheres attract each other (negative force) at sufficiently small distances if their voltage ratio is positive but not exactly equal to 1. For high size asymmetries (greater than 3:1), the repulsive force increases as the spheres separate from contact. In the "set voltage" case, at some finite distances, decreasing the voltage on a much larger second sphere (e.g. and ) starting from can increase the repulsion! Coulomb's law is obeyed when the spheres are at large separation.
References
[1] S. Banerjee, T. Peters, Y. Song and B. Wilkerson, "Closed-Form and Asymptotic Capacitance Coefficients for the Electrostatics of Two Spheres," Journal of Electrostatics, 101, 2019 103369. doi:10.1016/j.elstat.2019.103369.
[2] S. Banerjee, T. Peters, N. Brown and Y. Song, "Exact Closed-Form and Asymptotic Expressions for the Electrostatic Force between Two Conducting Spheres," Proceedings of the Royal Society A, 477(2246), 2021 A.4772020086620200866. doi:10.1098/rspa.2020.0866.
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