How To Calculate Coulomb's Law

Coulomb's law describes the force between two electrically charged particles that are stationary relative to each other. It is an inverse-square law, which means doubling the distance cuts the force to a quarter rather than half.

The constant k is 8.9875 billion in SI units, which is also written as 1 over 4 pi times the permittivity of free space. The large value exists because a coulomb is an enormous charge at the scale of individual particles.

Like charges repel and unlike charges attract. The sign of the product of the two charges tells you which: a positive product pushes them apart, a negative product pulls them together. The magnitude of the force is the same either way.

The law assumes point charges, or more precisely charges that are spherically symmetric. Real charge distributions need integration, but two small spheres behave exactly like point charges sitting at their centres.

Microcoulombs are the practical unit. One microcoulomb is one millionth of a coulomb, and typical static electricity demonstrations involve charges in the microcoulomb to nanocoulomb range.

The force falls off steeply. Halving the separation quadruples the force, and moving from 5 centimetres to 50 centimetres reduces it by a factor of a hundred. This is why electrostatic effects vanish so quickly at everyday distances.

Coulomb's law and Newton's law of gravitation have identical forms, both being inverse-square laws, but the constants differ enormously. Gravity is far weaker at particle scales but always attractive, so it dominates at large masses.

Comparing two electrons shows the scale. The electric repulsion between them outweighs their gravitational attraction by roughly forty two orders of magnitude, which is why gravity only shows up once large numbers of charges cancel out.

The medium between the charges matters. In a material rather than a vacuum, the force is reduced by the relative permittivity, so the same charges feel a weaker force in water than in air.

Superposition is the practical consequence. With three or more charges, each pair contributes its own Coulomb force, and the total force on any one charge is the vector sum of the pairwise forces. Direction has to be tracked alongside magnitude.

Coulomb's law was established with a torsion balance. Coulomb measured the twist of a suspended fibre to infer the force between charged pith balls, which is why the unit of charge carries his name.

The law is exact for point charges and for spherically symmetric ones, and a good approximation for anything else at distances large compared with the size of the objects. Only when the charges nearly touch does the simple form break down.

Field lines are the usual way to draw the result. Lines radiate outward from a positive charge and inward towards a negative one, and their density at any point is proportional to the force a small test charge would feel there.

The sign convention is easy to remember by thinking about energy. Like charges repel, so pushing them together takes work; unlike charges attract, so pulling them apart takes work. The force always points the way that releases stored energy.

Electrostatics underpins several technologies. Photocopiers and laser printers move charged powder with controlled fields, and electrostatic precipitators strip particles from industrial exhaust with exactly this force.