AP Physics C: Electricity and Magnetism

Charge

Charge is a fundamental property of matter, and it comes in two kinds, called positive and negative. It is a scalar, so it is added as an ordinary number with its sign, but the sign carries information: it decides whether two objects attract or repel. The SI unit is the coulomb (C).

Charge is quantised. The smallest free charge is the magnitude of the charge on an electron or a proton, the elementary charge

An electron has charge $-e$, a proton has $+e$ and a neutron has no charge. Any charge on an object is a whole-number multiple of $e$. Because $e$ is so small, the charges met in laboratory work (microcoulombs and nanocoulombs) contain enormous numbers of electrons, and treating charge as continuous is an excellent approximation when a distribution is being integrated over.

A point charge is a model in which the object's size is negligible compared with the distances in the problem. A charged sphere with its charge spread evenly behaves, outside its surface, as if all its charge were at its centre, so a sphere can be treated as a point charge when the other objects are outside it.

Coulomb's law

Two point charges $q_1$ and $q_2$ a distance $r$ apart exert forces on each other that are equal in size and opposite in direction, along the line joining them. The magnitude grows in direct proportion to each charge and falls with the square of the distance between them.

Here $\varepsilon_0 = 8.85\times10^{-12}\ \mathrm{C^2/(N\,m^2)}$ is the permittivity of free space and $k = 1/(4\pi\varepsilon_0) \approx 8.99 \times 10^9\ \mathrm{N\,m^2/C^2}$. Charges of the same sign repel and charges of opposite sign attract. Because the force is inversely proportional to $r^2$, doubling the separation cuts it to a quarter, and tripling the charge on one object triples the force.

In vector form, for the force on charge $q_2$ due to $q_1$,

where $\hat r$ is the unit vector pointing from $q_1$ towards $q_2$. With the signs kept in the product, a positive result means repulsion (the force on $q_2$ points away from $q_1$) and a negative result means attraction. Newton's third law holds exactly: the force on $q_1$ is the negative of this.

Several charges

The force on one charge from several others is the vector sum of the individual Coulomb forces, because each pair interacts as though the others were absent. This is the principle of superposition. The components are found separately and then added, so a sketch with the forces drawn on the charge being studied is the first step. A calculation with more than about four charges is unusual unless the arrangement is symmetric, in which case components often cancel.

Electric and gravitational forces

The two forces have the same inverse-square form, but they differ in important ways.

For two protons the electric repulsion is larger than their gravitational attraction by a factor of about $10^{36}$. Gravity nevertheless governs planets and galaxies, because large objects are almost exactly neutral, so their electric forces cancel while their gravitational forces, which never cancel, add up.

Everyday contact forces such as the normal force, friction and tension are in origin electric forces between the atoms of the surfaces. The number of particle interactions is so large that it is far more convenient to treat them as separate contact forces than to add Coulomb forces between individual atoms.

Worked examples

Two point charges

A charge $q_1 = +3.0\ \mu\mathrm{C}$ and a charge $q_2 = -2.0\ \mu\mathrm{C}$ are $0.30\ \mathrm{m}$ apart. The magnitude of the force on each is

The force is attractive, so each charge is pulled towards the other.

Three charges in a line

Charges $+1.0\ \mu\mathrm{C}$, $+2.0\ \mu\mathrm{C}$ and $-3.0\ \mu\mathrm{C}$ lie on the $x$ axis at $x = 0$, $x = 0.10\ \mathrm{m}$ and $x = 0.30\ \mathrm{m}$. To find the force on the middle charge, consider each of the others in turn.

The charge at $x = 0$ repels the middle charge in the $+x$ direction, with

The charge at $x = 0.30\ \mathrm{m}$ is $0.20\ \mathrm{m}$ away and is negative, so it attracts the middle charge in the $+x$ direction, with

Both forces point in $+x$, so the net force is $3.15\ \mathrm{N}$ in the $+x$ direction.

A charge in balance

Two identical small spheres of mass $m = 2.0\ \mathrm{g}$ hang from the same point on insulating threads of length $L = 0.50\ \mathrm{m}$ and repel each other until each thread makes a small angle $\theta$ with the vertical. For small angles the separation is $x \approx 2L\theta$, and the horizontal Coulomb force balances the horizontal component of tension, so $\tan\theta \approx \theta = F/mg$. With $F = kq^2/x^2$ this gives

For $\theta = 0.040\ \mathrm{rad}$, using $g = 9.8\ \mathrm{m/s^2}$,

The numerator is $4 \times 2.0\times10^{-3} \times 9.8 \times 0.25 \times 6.4\times10^{-5} = 1.254\times10^{-6}$, so $q^2 = 1.39\times10^{-16}\ \mathrm{C^2}$ and $q = 1.2\times10^{-8}\ \mathrm{C}$, about $12\ \mathrm{nC}$ on each sphere.