Potential Energy and Electric Potential

Electric potential is simply the energy each coulomb of charge would have at a given spot, and it's what we measure in volts. The name comes from potential energy, stored energy that has the potential to turn into motion, and it's the language engineers use for batteries, circuits and power lines. All we're really saying is that pushing a charge against an electric field is like lifting a book: it takes energy, the charge stores it, and it doesn't matter whether you go straight up or take a winding staircase. The formula $\Delta V = \Delta U/q$ is just two pieces: $\Delta U$ (the energy it took to move the charge) divided by $q$ (how much charge you moved), so a 9 V battery is a promise that every coulomb passing through it gains 9 J. The integral below looks heavier, but it's only "add up force times distance", so bear with it.

Why "potential" makes sense. A potential energy exists only if the work between two points doesn't depend on the path. Otherwise you could go up one way and down another and come out ahead, and "the energy at point B" wouldn't mean anything. The field of a point charge, $\frac{kq}{r^2}\hat{\mathbf{r}}$, is the gradient of $-\frac{kq}{r}$ (check: $\frac{d}{dr}\left(-\frac{kq}{r}\right) = \frac{kq}{r^2}$), so it's a conservative field, and line integrals of a gradient depend only on the endpoints. Every static charge distribution is a sum of point charges, so every electrostatic field is conservative, and we can define

along any path. Why the minus sign? Think of the book again. When you move with the field, the field does the work for you, like a book falling, so the potential drops. The work you do is the work integral of the force you apply, $-q\mathbf{E}$.

The point charge. Put the zero of potential at infinity and come in along a radial line: $V(r) = -\int_\infty^r\frac{kq}{u^2}\,du = \frac{kq}{r}$. Bringing a second charge $q_2$ in from infinity to distance $r$ takes work $q_2V = \frac{kq_1q_2}{r}$, the pair's potential energy. For a group of charges, add $\frac{kq_iq_j}{r_{ij}}$ over every pair.

Superposition without vectors. Here's where potential really pays off. The potential of many charges is $\sum\frac{kq_i}{r_i}$, a sum of signed numbers. Compare that with the last unit, where every field had to be split into components before you could add it. Numbers are far easier than vectors, which is why the next lesson gets $\mathbf{E}$ by finding $V$ first and differentiating.

Energy bookkeeping. With only electric forces acting, $K + qV$ stays constant, so $\Delta K = -q\,\Delta V$. Positive charges speed up as they fall to lower potential, like a ball rolling downhill. Negative charges do the opposite and speed up as they climb to higher potential, because for them $qV$ is upside down. One electron-volt is the energy an electron gains across 1 V, $1.60\times10^{-19}$ J. Chemical bonds are a few eV, visible photons about 2 eV, dental X-rays tens of keV, and medical linear accelerators deliver electrons at several MeV.

Uniform fields. Between parallel plates $V$ changes linearly, dropping by $Ed$ over a distance $d$ along the field. That's why field strength is quoted in volts per meter, the same unit as N/C.

A common mistake. Since $V$ and $U$ are both called "potential", it's tempting to treat them as the same thing. They aren't. $V$ is per coulomb, in volts, and belongs to the place; $U = qV$ is in joules and belongs to a particular charge sitting there. That's also where signs go wrong: an electron's potential energy is $-eV$, so it's lowest where $V$ is highest. A couple of other slips to watch for: the potential of a point charge is $kq/r$, not $kq/r^2$ (that's the field), and potentials add as plain signed numbers, never as vectors.

Voltage is how engineers talk about energy per charge. A USB line holds 5 V, an X-ray tube 30 to 150 kV, and a medical linear accelerator gives electrons several MeV to treat tumors. In a chip, the energy an electron gains across a transistor's fraction of a volt sets how much heat it dumps; in a solar cell, the band gap (1.1 eV for silicon) sets which photons can make current. The pair energy $kq_1q_2/r$ is the starting point for chemical bond energies and for the Coulomb barrier that fusion reactors must overcome.