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Coulomb's law

Electricity and magnetism · level 6, lesson 2 of 6

A question to start

Gravity pulls you down to the ground, yet a tiny balloon's charge can lift a hair against it. How strong is the force between charges?

The idea

Two point charges \(q_1\) and \(q_2\) a distance \(r\) apart push or pull with a force

\[F = \frac{k\,q_1 q_2}{r^2}, \qquad k = 8.99 \times 10^9 \text{ N m}^2/\text{C}^2.\]

This is Coulomb's law. Like charges repel, unlike attract. Like gravity, it is an inverse square law: double the distance and the force falls to a quarter.

The field of a single point charge follows from it: \(E = kq/r^2\).

Electric forces are enormous compared with gravity. Between a proton and an electron, the electric pull is about \(10^{39}\) times their gravitational pull.

F at rF/4 at 2rF/9 at 3rdistance

Worked examples

1. Two small charges. \(q_1 = q_2 = 1 \times 10^{-6}\) C, 10 cm apart: \(F = 8.99\times10^9 \times 10^{-12}/0.01 = 0.90\) N.

2. Inside hydrogen. A proton and an electron, \(1.60\times10^{-19}\) C each, \(5.29\times10^{-11}\) m apart: \(F = 8.99\times10^9 \times (1.60\times10^{-19})^2/(5.29\times10^{-11})^2 = 8.2\times10^{-8}\) N.

3. Further away. Move the two small charges to 30 cm: three times as far, so a ninth of the force, 0.10 N.

Try it

  1. Find the force between charges of \(3\times10^{-6}\) C and \(-2\times10^{-6}\) C, 0.2 m apart. Is it a push or a pull?
  2. What happens to the force if the distance is halved?
  3. Find the field 0.5 m from a charge of \(1\times10^{-6}\) C.
  4. Why do we notice gravity in everyday life far more than electric forces, if electric forces are so strong?
Answers
  1. \(8.99\times10^9 \times 6\times10^{-12}/0.04 = 1.35\) N; a pull, since the charges are unlike.
  2. It becomes four times as big.
  3. \(8.99\times10^9 \times 10^{-6}/0.25 = 36{,}000\) N/C.
  4. Most things have equal positive and negative charge, so their electric forces cancel. Gravity only adds up.

Common mistake

Forgetting to square the distance. \(r\) is squared: 0.1 m gives \(r^2 = 0.01\) m², not 0.1.

Where it leads