Key Points

The electrostatic potential energy difference between two points is the work done by an external force in moving a charge from one point to another without acceleration. It is given by $\Delta U = U_P - U_R = W_{RP}$.

Electrostatic potential (V) at a point is the work done per unit charge to bring a test charge from infinity to that point. For a point charge Q, the potential at a distance r is $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$. Its SI unit is the volt (V).

The potential due to an electric dipole at a large distance r is given by $V = \frac{1}{4\pi\epsilon_0} \frac{\mathbf{p} \cdot \hat{\mathbf{r}}}{r^2} = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2}$. Note that the potential falls off as $1/r^2$.

The electrostatic potential at any point due to a system of charges is the algebraic sum of the potentials due to individual charges. $V = V_1 + V_2 + \dots + V_n$.

An equipotential surface is a surface with a constant value of potential at all points. The electric field is always perpendicular to the equipotential surface, and no work is done in moving a charge on it.

The electric field is in the direction in which the potential decreases steepest. Its magnitude is given by the change in potential per unit displacement, $E = -\frac{dV}{dr}$.

The potential energy of a system of two charges $q_1$ and $q_2$ separated by a distance $r_{12}$ is the work done to assemble them. It is given by $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r_{12}}$.

The potential energy of a single charge q in an external potential $V(\mathbf{r})$ is $U = qV(\mathbf{r})$. For a dipole $\mathbf{p}$ in a uniform electric field $\mathbf{E}$, it is $U = -\mathbf{p} \cdot \mathbf{E}$.

In a static situation, the electrostatic field inside a conductor is zero, and the potential is constant throughout its volume. Any excess charge resides only on the outer surface of the conductor.

The electrostatic field at the surface of a charged conductor must be normal to the surface at every point. Its magnitude is given by $E = \frac{\sigma}{\epsilon_0}$, where $\sigma$ is the surface charge density.

A cavity inside a conductor is shielded from outside electric influences. The electric field inside the cavity is always zero, regardless of the charge on the conductor or external fields.

A capacitor is a system of two conductors separated by an insulator, used to store electric charge. Its capacitance (C) is the ratio of the charge (Q) on either conductor to the potential difference (V) between them, $C = \frac{Q}{V}$. The SI unit is the farad (F).

The capacitance of a parallel plate capacitor with plate area A and separation d, with vacuum between the plates, is given by $C = \frac{\epsilon_0 A}{d}$.

When a dielectric material is inserted between the plates of a capacitor, its capacitance increases by a factor K, known as the dielectric constant. The new capacitance is $C = K C_0$.

For capacitors connected in series, the reciprocal of the equivalent capacitance is the sum of the reciprocals of individual capacitances: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$. The charge on each capacitor is the same.

For capacitors connected in parallel, the equivalent capacitance is the sum of the individual capacitances: $C_{eq} = C_1 + C_2 + \dots + C_n$. The potential difference across each capacitor is the same.

The electrostatic energy (U) stored in a capacitor is given by the expressions $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$.

The energy stored in a capacitor can be viewed as being stored in the electric field between its plates. The energy per unit volume, or energy density (u), is given by $u = \frac{1}{2}\epsilon_0 E^2$.