Chapter 2 · Introduction

Chapter 2 — Electrostatic Potential and Capacitance

Charges fill space with an electric potential — a single number at every point that tells you the work per unit charge — and capacitors use that idea to store charge and energy.

The flash on a phone camera stores its energy for a fraction of a second in a small capacitor charged to about $300$ volts, then dumps it into the lamp in a thousandth of a second. A heart defibrillator does the same on a larger scale: a capacitor charged to a couple of thousand volts delivers a precisely measured pulse of energy through the chest. Your laptop keyboard, the touchscreen of your phone and the microphone in a recording studio all sense tiny changes in capacitance. And every time you plug in a charger, the word printed on it — volts — is a measure of electric potential, the central idea of this chapter.

In Chapter 1 we described the influence of charges by a vector, the electric field. Here we find a simpler description: a scalar — the electrostatic potential $V$ — that is just the work done per unit charge in bringing a test charge from far away. Potentials add as ordinary numbers, so they are far easier to calculate than fields, and the field can be recovered from them as the rate at which $V$ falls. Surfaces of constant potential show the field at a glance, and the same idea gives the energy stored in any arrangement of charges. Then we turn to matter: conductors, in which charge rearranges until the field inside vanishes, and dielectrics, which weaken a field by polarising. Put two conductors close together with a dielectric between them and you have a capacitor, a device that stores charge and energy — the last part of the chapter.

Work & potential→$V$ of charges & dipoles→Equipotentials, $E=-dV/dr$→Potential energy→Conductors & dielectrics→Capacitors & energy
🎯 By the end of this chapter you can…
  • define potential and potential difference, and calculate the potential of point charges, a dipole, systems of charges and a charged spherical shell;
  • sketch equipotential surfaces, explain why the field is perpendicular to them, and use $E=-dV/dr$ in both directions;
  • find the potential energy of a system of charges, of a charge in an external field and of a dipole in a uniform field;
  • state the electrostatic properties of conductors, explain polarisation of a dielectric and the meaning of the dielectric constant;
  • derive and use $C=\varepsilon_0A/d$, find the effect of a dielectric, combine capacitors in series and parallel, and calculate the energy stored;
  • solve board, JEE Main, JEE Advanced and NEET problems on all of the above.
Teaser — Riding the Potential Hill

The curve is the potential $V(x)$ along the line through the charges ($q=1$ nC, positions in cm), found simply by adding $kq_i/r_i$ for every charge. A small positive test charge (the yellow dot) is released on the line and drifts the way the field pushes it — always downhill on the $V$ curve. The field is the slope: $E=-dV/dx$. Click the track to release it somewhere else.

test charge at $x$—
potential $V$—
field $E_x=-dV/dx$—
energy $U=q_0V$ ($q_0=1$ nC)—

Sections in this chapter

2.1

Electric Potential due to Point Charges and Dipoles

Work in a conservative field, potential and potential difference, the potential of a point charge, a dipole, a system of charges and a charged shell.

$V=\dfrac{1}{4\pi\varepsilon_0}\dfrac{q}{r}$
Open section →2.2

Equipotential Surfaces and Electrostatic Potential Energy

Equipotentials, the field as the slope of the potential, energy of a system of charges and of a dipole in a field.

$E=-\dfrac{dV}{dr}$, $\ U=\dfrac{kq_1q_2}{r_{12}}$
Open section →2.3

Conductors, Dielectrics and the Parallel-Plate Capacitor

Why $E=0$ inside a conductor, polarisation and the dielectric constant, capacitance and the parallel-plate capacitor.

$C=\dfrac{K\varepsilon_0A}{d}$
Open section →2.4

Combinations of Capacitors and Stored Energy

Series and parallel combinations, energy stored in a capacitor, energy density and sharing of charge.

$U=\dfrac{Q^2}{2C}=\tfrac12CV^2$
Open section →
✅ Before you start
This chapter builds directly on Chapter 1: Coulomb's law and superposition (Section 1.1), the electric field, the dipole and the torque on a dipole (Section 1.2), and Gauss's law with the fields of a plane sheet and a spherical shell (Section 1.3). From Class 11 Physics you need work done by a force, kinetic and potential energy, and the idea of a conservative force (Class 11 Physics, Chapter 5 — Work, Energy and Power). From mathematics you need derivatives and the simplest integral $\int r^{-2}\,dr=-r^{-1}+C$; every section says where integration is used.
← 1.3 Electric Flux, Gauss's Law and Its Applications2.1 Electric Potential due to Point Charges and Dipoles →