Electric Force, Field, and Potential

Charge is a basic property of matter: a proton carries +e, an electron −e, and e is treated as the smallest amount of charge. Coulomb's law, $F_E = \frac{k\lvert q_1q_2\rvert}{r^2}$, gives the force between point charges, which is far stronger than gravity between them; a material's permittivity describes how easily its charges shift, or polarize, in an electric field.

An object's net charge changes only when charge, usually electrons, moves onto or off it. Rubbing and touching transfer charge, a nearby charged object can polarize a neutral one without touching it, and grounding connects an object to Earth so charge can flow on or off; charging by induction combines the last two.

An electric field is the force per unit charge a small test charge would feel, $\vec{E} = \frac{\vec{F}_E}{q}$; it points away from positive charges and toward negative ones, and fields from several charges add as vectors. In a conductor at equilibrium, extra charge sits on the surface, the field inside is zero, and the field just outside is perpendicular to the surface.

Electric potential energy is stored in the arrangement of charges: for two point charges $U_E = \frac{kq_1q_2}{r}$, the work needed to bring them together from infinitely far apart. It's a scalar, so for several charges you add up the energy of every pair; it's positive for like charges and negative for opposite ones.

Electric potential is potential energy per unit charge, measured in volts; near point charges you add up $V = \frac{kq}{r}$ from each one as plain numbers. Equipotential lines connect points at the same potential, the field crosses them at right angles and points toward lower potential, and the average field between two points is $E = \left\lvert \frac{\Delta V}{\Delta r} \right\rvert$.

A parallel-plate capacitor is two conducting plates holding equal and opposite charges, with capacitance $C = \frac{Q}{\Delta V} = \frac{\kappa\varepsilon_0 A}{d}$. The field between the plates is uniform, so a charge there accelerates steadily like a projectile; the stored energy is $U_C = \frac{1}{2}Q\Delta V$, and putting a dielectric between the plates raises the capacitance.

When a charge moves through a potential difference, the electric potential energy changes by $\Delta U_E = q\Delta V$, and by conservation of energy the kinetic energy changes by the opposite amount. That's how a few volts can get an electron moving fast, and why a positive charge let go in a field speeds up toward lower potential.