Assembling a Charge Configuration

Bring charges in from infinity one at a time and keep a running tab. The first charge costs nothing to place, since no field exists yet to fight. The second charge costs kq1q2/r12, the work to push it through the first charge's field. The third charge costs the work against both charges already sitting there, and so on. Each new arrival faces a bigger combined field than the last, so the marginal cost of assembly keeps climbing. Add up all the pairwise terms and you get the total potential energy of the configuration: the work it took to build it, and the energy you would get back if it fell apart.

For a continuous object like a uniformly charged sphere, you cannot track individual charges, so you build it up in thin shells instead. At some stage of construction the sphere has radius r and charge q already in place. Bringing in the next thin shell of charge dq from infinity costs dU = k q dq / r. Writing q and dq in terms of r using the sphere's constant charge density, then integrating from r = 0 to r = R, gives U = (3/5) kQ²/R. The 3/5 factor is the fingerprint of a uniform solid sphere; a different charge distribution gives a different number out front.

This assembly energy is exactly the potential energy that gets released if the configuration is allowed to fly apart, turning into kinetic energy of the receding charges. That mirrors a stretched spring: winding it up stores energy equal to the work done against its restoring force, and letting it snap back converts that stored energy into motion. The next step in the unit flips the perspective, using potential as a single scalar function of position rather than tallying work charge by charge.