Electric Potential

The electric potential energy of two point charges is $U_E = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r}$, taking zero at infinite separation; it equals the work an outside force does to bring them together from far away. It’s positive for like charges and negative for opposite charges, and for a group of charges you add up the energy of every pair.

Electric potential is potential energy per unit charge, measured in volts; for a point charge $V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}$, and potentials from several charges simply add as numbers (integrate for a continuous distribution). Potential and field are linked by $\Delta V = -\int \vec{E}\cdot d\vec{r}$ and $E_x = -\frac{dV}{dx}$, so the field points toward lower potential and crosses equipotential lines at right angles.

When a charge q moves through a potential difference $\Delta V$, the potential energy of the system changes by $\Delta U_E = q\Delta V$, and if no other forces do work that change shows up as kinetic energy, $\Delta K = -q\Delta V$. Positive charges speed up as they move toward lower potential, while negative charges such as electrons speed up moving toward higher potential.