Work and Electric Potential Energy

Push two positive charges toward each other and the repulsion fights you the whole way, like compressing a spring. That effort doesn't vanish — it gets stored in the configuration itself as electric potential energy, ready to be released as kinetic energy if you let go. The energy lives in the arrangement of charges, not in either charge alone.

The mechanism is just the work-energy idea applied to Coulomb's force. Define potential energy as zero when charges sit infinitely far apart. Then U = kq1q2/r is the work an outside agent must do to bring charge q2 from infinity to a distance r from fixed charge q1, moving it slowly so kinetic energy stays zero. Same-sign charges give positive U — you had to push against repulsion, and that energy is recoverable as the charges fly apart. Opposite-sign charges give negative U — the charges wanted to come together, attraction did the work for you, so you'd have to add energy to pull them back apart.

Because the Coulomb force is conservative, this work depends only on the final separation r, never on the route taken to get there. That's the same property gravity has, and it's why U can be defined as a function of position at all. This sets up the next idea: dividing U by the test charge gives electric potential V, a scalar field that lets you skip vector force arguments entirely and just subtract numbers to get energy changes.