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For a collection of fixed electrical conductors in linear electrostatics, the capacitance matrix relates conductor charges to potentials relative to a specified reference: $Q_i={\sum }_j{C}_{ij}{V}_j$. It includes mutual electrostatic effects omitted when each conductor is treated as an isolated capacitor. Its electrostatic energy is $U=\frac{1}{2}{\mathbf{V}}^{\mathsf{T}}\mathbf{CV}=\frac{1}{2}{\sum }_i{Q}_i{V}_i$. Geometry and separation are needed to determine its entries.

For charge sharing between distant connected spheres, use the standard approximation that the spheres are sufficiently far apart for mutual electrostatic effects to be negligible, and that the wire adds negligible capacitance. The capacitance of an isolated conducting sphere is $C=4\pi {ϵ}_{0}r$. ThusThe initial charges are $Q_i=C_i{V}_i$. Connecting the spheres allows charge to flow until their potentials are equal. Charge conservation givesThe initial and final electrostatic energies are $U_i=\frac{1}{2}{C}_1{V}_1^2+\frac{1}{2}{C}_2{V}_2^2$ and $U_f=\frac{1}{2}(C_1+C_2)V_f^2$. Their difference becomes Joule heating in the resistive wire:Numerically,The resistance determines how fast this happens, not the final energy difference. Sufficient damping suppresses sustained charge-current oscillations; the calculation also neglects energy radiated away. At finite sphere separation the capacitance matrix would be needed, and the radii and initial potentials alone would not specify an exact correction.