Theory

Electric charge and Coulomb's law

What electric charge is and what properties it has, how conductors and insulators are charged, how Coulomb's law and the superposition principle give the force between point charges in one and two dimensions, and when a system of charges is in equilibrium.

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Electric charge
Conductors and insulators

Electric charge

Throughout the topic the charges are at rest in an inertial reference frame and are in vacuum, except where stated otherwise.

Electric charge is the scalar quantity that measures the ability of a body to exert and experience electric forces. There are two kinds of charge, conventionally called positive and negative; two bodies with charges of the same sign repel each other, two bodies with charges of opposite sign attract each other. The charge of a body is the algebraic sum of the charges of its constituents, and a body with zero total charge is called neutral.

Ordinary matter is made of atoms, with a nucleus of protons and neutrons surrounded by electrons. The proton has charge $+ e$ , the electron has charge $- e$ and the neutron is neutral, with

e = 1.60 \cdot 1 0^{- 19} C .

A body is negatively charged if it has more electrons than protons, positively charged if it has fewer. The unit in the International System is the coulomb, related to the ampere, the unit of current, by $1 C = 1 A \cdot s$ ; for current we refer to the topic on electric current and direct-current circuits. The charges of laboratory bodies are much smaller than one coulomb, and the submultiples $1 μ C = 1 0^{- 6} C$ and $1 nC = 1 0^{- 9} C$ are used.

1. Quantization of charge

It is an experimental fact, first measured accurately by Robert Millikan with the oil-drop experiment, that the charge of every isolated body is an integer multiple of the elementary charge:

q = n e, n \in Z .

Quarks, the constituents of protons and neutrons, have charges $\pm \frac{1}{3} e$ and $\pm \frac{2}{3} e$ , and are not observed in isolation; therefore measurable charges are multiples of $e$ .

The law makes it possible to count the elementary charges transferred in a charging process. A body with charge $q = - 3.2 nC$ has

n = \frac{∣ q ∣}{e} = \frac{3.2 \cdot 1 0 ^{- 9}}{1.60 \cdot 1 0 ^{- 19}} = 2.0 \cdot 1 0^{10}

excess electrons. Since $e$ is very small, in macroscopic bodies $n$ is enormous, and charge is treated as a quantity that varies continuously.

2. Conservation of charge

It is an experimental principle, verified in all known processes, including reactions between elementary particles, according to which in an isolated system the algebraic sum of the charges remains constant in time,

i \sum q_{i} = constant .

Charge moves from one body to another and its algebraic sum stays the same; new charges appear only in pairs of opposite sign, as in the creation of an electron and a positron from a high-energy photon.

The principle fixes the total charge in contact and rubbing problems. If two bodies with charges $+ 5.0 nC$ and $- 1.0 nC$ touch and are then separated, the sum of their final charges is $+ 4.0 nC$ , whatever the distribution.

Conductors and insulators

A conductor is a material in which some of the charges, called conduction charges, move freely within it; in metals they are the outermost electrons of the atoms, not bound to a single nucleus. An insulator, or dielectric, is a material in which every electron remains bound to its own atom or molecule. Metals, graphite, salt solutions and the human body are conductors; glass, plastic, rubber, ceramic and dry air are insulators. Semiconductors, such as silicon, have an intermediate behavior, which depends on temperature and impurities.

The distinction determines where the charge of a charged body is located. On an insulator the charge remains in the region where it was deposited. On a conductor the excess charges repel each other and move until they reach an equilibrium configuration, in which they lie on the outer surface; for the properties of conductors in equilibrium we refer to the topic on conductors, capacitors and dielectrics.

Grounding

The Earth is a very large conductor, which can give up or absorb electrons without its electrical state changing appreciably. Connecting a conductor to ground with a metal wire allows the conduction charges to pass between the conductor and the Earth; therefore a grounded conductor, far from other charged bodies, discharges. Grounding is used in charging by induction, and in electrical installations it prevents the metal casings of appliances from remaining charged.

The theory continues with 6 more sections (charging, coulombs law, superposition principle…), plus all worked examples, the solutions and the PDF.

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