Coulomb’s Law
Electrostatic force between two point charges, with unit conversion built in.
q₁ = 2.0000e-6 C, q₂ = -3.0000e-6 C, r = 0.05 m
Coulomb’s law gives the force between two point charges: F = k·q₁q₂ ÷ r², with k = 8.988×10⁹ N·m²/C². Charges of +2 µC and −3 µC held 5 cm apart attract with a force of about 21.6 N — opposite signs attract, like signs repel.
The same shape as gravity
Coulomb’s law and Newton’s law of gravitation are built identically: a constant, the product of two source quantities on top, and the square of the separation underneath. Charge takes the place of mass, and k takes the place of G. Recognising that shape means the intuition transfers — double the separation and the force drops to a quarter in both cases.
One difference matters enormously. Mass comes in only one sign, so gravity only attracts. Charge comes in two, so the electrostatic force can push as well as pull, and the sign of the product q₁q₂ is what decides. The other difference is scale: k is about 10²⁰ times larger relative to typical charges than G is to typical masses, which is why static electricity can lift your hair against the pull of the entire Earth.
Point charges, and what that assumes
The law describes charges small enough to treat as points, with r measured centre to centre. For uniformly charged spheres it still works exactly, using the centres. For irregular or extended objects the field has to be integrated instead, and for charges close enough that they redistribute each other’s surface charge the point approximation quietly stops being accurate.
k = 8.988×10⁹ N·m²/C²; charges in coulombs, separation in metres, force in newtons
- 1 Convert both charges to coulombs. 2 µC is 2×10⁻⁶ C and −3 µC is −3×10⁻⁶ C.
- 2 Convert the separation to metres. 5 cm is 0.05 m — leaving it in centimetres makes the answer wrong by a factor of 10 000.
- 3 Square the separation. 0.05² = 0.0025 m².
- 4 Multiply out. 8.988×10⁹ × (2×10⁻⁶ × 3×10⁻⁶) ÷ 0.0025 = 21.57 N.
- 5 Use the signs to name the direction. The product q₁q₂ is negative, so the charges attract; a positive product would repel.
How the force changes with separation
Two 1 µC charges. The inverse square means small changes in distance matter a lot.
| Separation | Force |
|---|---|
| 1 cm | 89.88 N |
| 2 cm | 22.47 N |
| 5 cm | 3.595 N |
| 10 cm | 0.8988 N |
| 20 cm | 0.2247 N |
| 1 m | 0.008988 N |
Where the arithmetic goes wrong
Unit conversion causes most errors here, and the inverse square amplifies them. A separation left in centimetres is out by 10⁴; a charge left in microcoulombs is out by 10⁶ for each charge, so 10¹² in total. Convert everything to coulombs and metres before anything else.
The second trap is the sign. Substituting signed charges gives a signed result, and that sign carries the meaning: negative for attraction, positive for repulsion. But the force magnitude uses the absolute value, and a negative number in a magnitude answer is not meaningful. Work with magnitudes and state the direction separately. Finally, forces are vectors — with three or more charges, each pair contributes a force and the total is the vector sum, not the arithmetic one.