Coulomb's law and the electric force

Electric charge

Electricity begins with electric charge. Charge is a fundamental property of matter, just as mass is. There are two kinds of electric charge, called positive and negative. Like charges repel, and opposite charges attract.

Charge is measured in coulombs, written $C$. The elementary charge has magnitude

$e=1.602 imes 10^{-19} C$

A proton has charge $+e$, and an electron has charge $-e$. Ordinary objects often contain enormous numbers of positive and negative charges, but they usually appear neutral because the amounts balance.

Charge is conserved. In an isolated system, the total electric charge does not change. Charges may move from one object to another, but they are not created or destroyed in ordinary processes.

Coulomb's law

For two point charges $q_1$ and $q_2$ separated by distance $r$, Coulomb's law gives the magnitude of the electric force:

F=k_erac{|q_1q_2|}{r^2}

The constant $k_e$ is Coulomb's constant:

k_e=rac{1}{4piepsilon_0}

where $epsilon_0$ is the permittivity of free space. Numerically,

$k_eapprox 8.99 imes 10^9 N,m^2/C^2$

The force points along the line connecting the charges. If the charges have the same sign, the force is repulsive. If they have opposite signs, the force is attractive.

Vector form

The vector form of Coulomb's law is especially important in three-dimensional problems. The force on charge 2 due to charge 1 may be written

ec{F}*{2leftarrow 1}=k_erac{q_1q_2}{r^2}hat{r}*{2leftarrow 1}

where $hat{r}_{2leftarrow 1}$ is a unit vector pointing from charge 1 toward charge 2. The sign of $q_1q_2$ determines whether the force points along or opposite this unit vector.

Vector notation prevents confusion when charges are not arranged on a single line.

Superposition of forces

Electric forces obey superposition. If several charges act on a charge $q$, the net force is the vector sum of the individual forces:

ec{F}_{net}=sum_i ec{F}_i

Each force is calculated as though the other charges were absent, then the vector forces are added.

This principle is powerful but requires careful geometry. In two or three dimensions, components are usually the safest method:

$F_x=sum_i F_{ix}$

$F_y=sum_i F_{iy}$

and similarly for $z$.

Comparison with gravity

Coulomb's law resembles Newton's law of gravitation:

F_g=Grac{m_1m_2}{r^2}

Both are inverse-square laws. Both act along the line joining two objects. Both can be described by fields.

But there are major differences. Gravity is always attractive, while electric force can attract or repel. Electric force is also enormously stronger than gravity at the particle scale. The reason ordinary matter is not dominated by huge electric forces is that positive and negative charges often nearly cancel.

Point charges and idealization

Coulomb's law in its simplest form applies to point charges, meaning charges whose size is negligible compared with their separation. Real charged objects can often be approximated as point charges if observed from far away.

For extended charge distributions, the total force can be found by adding contributions from small charge elements. In calculus form, this becomes an integral.

Units and signs

Always keep signs separate from magnitudes. The magnitude formula uses $|q_1q_2|$. Direction must be determined physically or by vector form. A positive force value in one-dimensional notation depends on your chosen coordinate axis; it does not automatically mean repulsion.

The big idea

Coulomb's law is the starting point of electrostatics. It says electric charges exert inverse-square forces on one another, with direction determined by charge signs and geometry. Together with superposition and vector addition, Coulomb's law allows the force from any static arrangement of point charges to be calculated in principle.