Electrostatic Potential and Capacitance

This chapter introduces the scalar concepts of electrostatic potential and potential energy, which offer an alternative, often simpler, energy-based approach to problems that were previously solved using the vector quantities of electric force and field. The first part of the chapter defines potential difference, calculates the potential due to various charge distributions like point charges and electric dipoles, and explores the relationship between electric field and potential. You will learn to visualize potential using equipotential surfaces and calculate the potential energy stored in a system of charges.

The second half focuses on capacitance, the ability of a system to store electric charge and energy. It details the construction and working of capacitors, with a primary focus on the parallel plate capacitor. You will learn to calculate capacitance and see how it is affected by the geometry of the capacitor and the introduction of dielectric materials between its plates. The chapter also covers the analysis of capacitor circuits with series and parallel combinations, and the calculation of energy stored in a capacitor.

This chapter relies heavily on concepts from the previous chapter, "Electric Charges and Fields," so a firm grasp of Coulomb's law, electric field, and Gauss's law is essential. The shift from vector fields to scalar potentials can be conceptually tricky. Difficult problems often involve complex capacitor networks, charge redistribution when capacitors are connected or disconnected, and the effects of inserting dielectric slabs under different conditions (battery connected vs. disconnected).

Formulas for Electrostatic Potential and Capacitance

Electrostatic potential due to a point charge: $V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$ V is the potential at a distance r from a point charge q, assuming V=0 at infinity.
Relation between electric field and potential: $E = -\frac{dV}{dr}$ The electric field is the negative of the potential gradient. Holds for spherically symmetric fields.
Potential energy of a two-charge system: $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r}$ The potential energy of two point charges q1 and q2 separated by distance r.
Potential energy of a dipole in an external field: $U = -\vec{p} \cdot \vec{E}$ p is the dipole moment and E is the uniform external electric field.
Definition of capacitance: $C = \frac{Q}{V}$ Q is the magnitude of charge on one plate and V is the potential difference between them.
Capacitance of a parallel plate capacitor: $C = \frac{K \epsilon_0 A}{d}$ A is the plate area, d is the separation, and K is the dielectric constant (K=1 for vacuum).
Capacitors in series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$ The charge on each capacitor is the same.
Capacitors in parallel: $C_{eq} = C_1 + C_2 + \dots$ The potential difference across each capacitor is the same.
Energy stored in a capacitor: $U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$ This energy is stored in the electric field between the plates.
Energy density of an electric field: $u_E = \frac{1}{2} \epsilon_0 E^2$ Energy stored per unit volume in the electric field in vacuum.