Gauss's law relates the electric flux through any closed surface (a Gaussian surface) to the charge enclosed by it:
$$\oint \vec{E}\cdot d\vec{A}=\frac{q_{enc}}{\epsilon_0}$$ It holds for any closed surface and any charge distribution, but it becomes a powerful tool for finding fields only when the symmetry lets us take $E$ out of the integral.
Three standard applications follow directly:
- Infinite line charge of linear density $\lambda$: $E=\frac{\lambda}{2\pi\epsilon_0 r}$ (field falls as $1/r$, directed radially).
- Infinite charged sheet of surface density $\sigma$: $E=\frac{\sigma}{2\epsilon_0}$ (uniform, independent of distance).
- Charged spherical shell of charge $Q$: outside, $E=\frac{1}{4\pi\epsilon_0}\frac{Q}{r^2}$ (as if all charge were at the centre); inside, $E=0$.
The electric potential $V$ at a point is the work done per unit positive charge in bringing it from infinity to that point: $V=\frac{W}{q}$, measured in volts (V = J/C). For a point charge,
$$V=\frac{1}{4\pi\epsilon_0}\frac{q}{r}$$ Potential is a scalar, so potentials due to many charges simply add algebraically. The potential of a dipole at distance $r$ (for $r\gg a$) is $V=\frac{1}{4\pi\epsilon_0}\frac{p\cos\theta}{r^2}$ — zero on the equatorial line ($\theta=90^\circ$).
Equipotential surfaces are surfaces on which $V$ is constant. No work is done in moving a charge along them, so the electric field is always perpendicular to an equipotential surface. For a point charge they are concentric spheres; for a uniform field they are parallel planes.
Field and potential are linked by $$E=-\frac{dV}{dr}$$ The field points in the direction of decreasing potential, and its magnitude equals the potential gradient. The potential energy of a system of two charges is $U=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r}$; for more charges, sum over every distinct pair. A positive $U$ means work was done to assemble the (repelling) system; a negative $U$ means the configuration is bound.