All electrical phenomena begin with charge: lightning, hair sticking to a comb, chemical bonds, electric current. The force between charges, Coulomb’s law, looks like the law of universal gravitation, but it can attract or repel and is incomparably stronger than gravity. This topic is the foundation of electrostatics: the electric field, potential and capacitors all start here.
§1. Electric charge and its properties
Electric charge is the physical quantity that expresses the ability of particles and bodies to take part in electromagnetic interaction. Its unit is the coulomb (C): 1 C is the charge passing through a conductor’s cross-section in 1 s at a current of 1 A.
There are two kinds of charge, positive and negative. Like charges repel; unlike charges attract.
Charge is quantised: every charge is a whole multiple of the elementary charge:
q = N·e, e = 1.6·10⁻¹⁹ C
The electron carries −e, the proton +e, the neutron no charge. An ordinary atom has equal numbers of protons and electrons, so it is neutral.
A body that loses electrons becomes positive; one that gains electrons becomes negative. Protons are tightly bound in nuclei, so usually only electrons move between bodies.
Charge does not depend on a body’s speed and is the same in every frame of reference.
1 C is a huge charge: it corresponds to 1/(1.6·10⁻¹⁹) ≈ 6.25·10¹⁸ electrons. That is why problems usually give charges in microcoulombs (1 μC = 10⁻⁶ C) and nanocoulombs (1 nC = 10⁻⁹ C).
§2. Charging bodies
There are several ways to charge a body:
Friction: when an ebonite rod is rubbed with wool, electrons pass from the wool to the ebonite; the ebonite becomes negative and the wool positive. Glass rubbed with silk becomes positive. Both bodies receive equal and opposite charges.
Contact: when a charged body touches an uncharged conductor, part of the charge moves onto it, and both end up with charge of the same sign.
Induction: when a charged body is brought near a conductor, the free electrons in it redistribute; the near end gets an opposite charge and the far end a like charge. If the far end is earthed and then disconnected, the conductor stays charged.
The attraction of uncharged scraps of paper to a charged comb is explained the same way: the paper is polarised, and the nearer, opposite charge is attracted more strongly.
Charge is detected with an electroscope or an electrometer: when charged, the like-charged leaves (or the needle) repel and spread apart.
§3. Conservation of charge
In a closed (isolated) system the algebraic sum of charges stays constant: q₁ + q₂ + ... = const
Charge is neither created from nothing nor destroyed; it is only redistributed between bodies. Even in rubbing, charge is not “made”: as many electrons as one body loses, the other gains.
When identical conducting spheres touch, the total charge is shared equally:
q = (q₁ + q₂)/2 (with signs)
Worked example
Identical metal spheres with charges +6 nC and −2 nC are touched together and separated. What is the charge on each, and how many electrons moved?
Total charge: 6 + (−2) = 4 nC; each gets q = 4/2 = 2 nC.
The second sphere went from −2 nC to +2 nC, so it lost electrons carrying 4 nC.
N = 4·10⁻⁹/(1.6·10⁻¹⁹) = 2.5·10¹⁰ electrons moved from the second sphere to the first.
§4. Coulomb’s law
The French physicist Charles Coulomb (1785) used a torsion balance to measure the force between point charges:
F = k·|q₁|·|q₂| / (εr²), k = 9·10⁹ N·m²/C²
A point charge is a charged body much smaller than the distance between the bodies.
The force acts along the line joining the charges; by Newton’s third law both charges feel equal and opposite forces.
ε is the relative permittivity of the medium (vacuum, air: 1; kerosene: 2; water: 81). A medium reduces the force ε times.
k = 1/(4πε₀), ε₀ = 8.85·10⁻¹² F/m is the electric constant.
The force is inversely proportional to the square of distance: tripling the distance makes the force 9 times smaller.
Worked example
Two charges of 2 μC each are 30 cm apart in air. What force acts between them?
Convert to SI: q = 2·10⁻⁶ C, r = 0.3 m.
F = 9·10⁹·(2·10⁻⁶)²/0.3² = 9·10⁹·4·10⁻¹²/0.09.
F = 0.036/0.09 = 0.4 N (repulsion).
§5. Superposition of forces and comparison with gravity
If several charges act on one charge, each force is found separately from Coulomb’s law and the forces are added as vectors (the superposition principle). For charges on one line the forces add or subtract depending on direction.
The electric force is incomparably stronger than gravity. In a hydrogen atom the Coulomb force between electron and proton is ≈ 8.2·10⁻⁸ N, while the gravitational force is ≈ 3.6·10⁻⁴⁷ N, a ratio of order 10³⁹. Large bodies are usually neutral, so gravity dominates on cosmic scales, while electric forces rule inside atoms and molecules.
Worked example
Charges +1 μC and +4 μC are 30 cm apart. Where will a third charge be in equilibrium?
Both charges are positive, so the point lies between them: distance x from the first and 0.3 − x from the second.
0.3 − x = 2x → x = 0.1 m, 10 cm from the smaller charge (the size and sign of the third charge do not matter).
§6. Interaction energy and work
The potential energy of two point charges is
W = k·q₁q₂/(εr)
For like charges W > 0, for unlike charges W < 0. Pushing like charges together, an external force does positive work, which is stored as potential energy:
A_ext = W₂ − W₁ = kq₁q₂(1/r₂ − 1/r₁)
Worked example
Two charges of 3·10⁻⁸ C in air are brought from 50 cm to 10 cm apart. How much work does the external force do?