Coulomb’s law states that the electrostatic force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance between them, acting along the line joining them: F = k × |q₁q₂| / r².
Learning objectives
- State Coulomb’s law and identify each variable.
- Calculate the magnitude of the force between two point charges.
- Determine whether the force is attractive or repulsive from the signs.
- Explain superposition and the limits of the point-charge model.
Progressive explanation
Start with the plain-language idea, then move into the engineering detail. Each level stands on its own, so you can stop at the depth you need.
Beginner The core idea
Like charges repel and opposite charges attract. If you double one charge, the force doubles; if you double the distance, the force falls to one quarter. The constant k is very large, which is why ordinary static charges produce noticeable forces even though the charge in a single electron is tiny.
Intermediate How it behaves in a circuit
For more than two charges, the total force on one charge is the vector sum of the individual forces from all the others, a principle called superposition. Coulomb’s law applies exactly to point charges in vacuum. For extended charged objects, the force is found by integrating over the charge distribution, and for charged spheres outside their surfaces the result is the same as if all the charge were at the centre.
Advanced Limits, assumptions, engineering detail
The inverse-square law is tested to high precision and is linked to the zero mass of the photon. In engineering, Coulomb’s law explains contact potential, electrostatic precipitators, electrophotography and the forces in MEMS devices. It is also the starting point for deriving the electric field of a charge distribution and for Gauss’s law, which is often more convenient for symmetric geometries.
Formulas and variables
Units of measurement
| Quantity | Symbol | Unit |
|---|---|---|
| Electric force | F | newton (N) |
| Electric charge | q | coulomb (C) |
| Distance | r | metre (m) |
| Permittivity of free space | ε₀ | farad per metre (F/m) |
Worked examples
Force between two microcoulomb charges
Problem. Two point charges of 2.0 µC and 3.0 µC are 0.30 m apart in air. Find the force between them.
Solution. F = k × |q₁q₂| / r² = 8.99 × 10⁹ × (2.0 × 10⁻⁶ × 3.0 × 10⁻⁶) / (0.30)² = 8.99 × 10⁹ × 6.0 × 10⁻¹² / 0.090 ≈ 0.599 N.
Why one coulomb is a large charge
Problem. Estimate the force between two 1.0 C charges separated by 1.0 m.
Solution. F = 8.99 × 10⁹ × (1.0 × 1.0) / 1.0² = 8.99 × 10⁹ N.
Practical applications
- Electrostatic precipitators and dust collection
- Electrophotography and laser printing
- MEMS and electrostatic actuators
- Understanding contact electrification and ESD
Automotive applications
Static charge on a vehicle body can produce a noticeable shock when you touch the door handle, and the same charge can damage electronic components during service. The force described by Coulomb’s law is small for the charge involved, but the voltage can be high. Anti-static straps and proper grounding during repair reduce both the shock and the risk to electronics.
Industrial applications
Electrostatic precipitators use the force on charged particles to remove dust from industrial exhaust. Powder coating uses charged paint particles attracted to a grounded workpiece, giving an even coating. In both cases, the force follows Coulomb’s law, and the design of electrodes and field distribution determines collection or coating efficiency.
Measurement procedure
- The electrostatic force is usually calculated from charge and geometry rather than measured directly; a torsion balance or sensitive force sensor can measure it in laboratory conditions.
- Charge can be measured with an electrometer or calculated from current and time, then used with Coulomb’s law.
- The inverse-square relation can be checked by measuring force at several distances with a calibrated setup.
- In air, humidity affects charge leakage, so electrostatic measurements require controlled conditions.
Common mistakes
- Forgetting to square the distance when applying the inverse-square law.
- Using centimetres or millimetres in the formula instead of metres.
- Thinking the sign of the charges changes the magnitude of the force; it only changes attraction versus repulsion.
- Applying the point-charge formula to large charged objects without checking that the charges can be treated as concentrated at their centres.
Safety considerations
- Static discharges can ignite flammable vapours; bond and earth equipment during fuel transfer.
- High-voltage electrostatic equipment can store dangerous energy in capacitors even after power is removed.
- ESD can destroy electronic components; use grounded wrist straps and ESD-safe packaging.