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.

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

.

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.

[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.