How strongly do two charges push or pull? In 1785 Charles-Augustin de Coulomb measured it with a delicate twisting balance. The answer looks just like Newton's law of gravitation, with charges in place of masses. It builds on the charges you just met.
Why it works
The force between two point charges grows in proportion to each charge and falls off as the square of the distance. The inverse square has a picture behind it: the influence of a charge spreads out over spheres whose area grows as $4\pi r^2$, so at twice the distance it is spread over four times the area. Unlike gravity, charge comes in two signs, so the force can push as well as pull. And it is enormously stronger: between an electron and a proton the electric force is about $10^{39}$ times the gravitational one.
Key idea
$$F = k\frac{|q_1 q_2|}{r^2}, \quad k = 8.99 \times 10^{9}\,\text{N}\cdot\text{m}^2/\text{C}^2$$ The force acts along the line between the charges: like charges repel, unlike attract. Each charge feels the same size of force (Newton's third law).
Worked examples
Worked example: Two charges
Find the force between $+3.0\,\mu\text{C}$ and $-5.0\,\mu\text{C}$ charges $30\,\text{cm}$ apart.
- SI units: $3.0 \times 10^{-6}\,\text{C}$, $5.0 \times 10^{-6}\,\text{C}$, $r = 0.30\,\text{m}$.
- $$F = (8.99 \times 10^{9})\dfrac{(3.0 \times 10^{-6})(5.0 \times 10^{-6})}{(0.30)^2} = \dfrac{0.135}{0.090} = 1.50\,\text{N}.$$
- Opposite signs, so the force is attractive: each charge is pulled toward the other with $1.50\,\text{N}$.
Answer: $1.50\,\text{N}$, attractive
Worked example: Scaling
Two charges attract with $12\,\text{N}$. One charge is doubled and the distance is tripled. What is the new force?
- Doubling a charge doubles $F$. Tripling $r$ divides $F$ by $3^2 = 9$.
- $F = 12\,\text{N} \times \dfrac{2}{9} = 2.67\,\text{N}$.
Answer: $2.67\,\text{N}$
Worked example: Inside a hydrogen atom
An electron and a proton are $5.29 \times 10^{-11}\,\text{m}$ apart. Compare the electric force with the gravitational force ($6.67 \times 10^{-11}\,\text{N}\cdot\text{m}^2/\text{kg}^2$).
- Electric: $$F_e = \dfrac{(8.99 \times 10^{9})(1.60 \times 10^{-19})^2}{(5.29 \times 10^{-11})^2} = 8.22 \times 10^{-8}\,\text{N}.$$
- Gravity: $$F_g = \dfrac{(6.67 \times 10^{-11})(9.11 \times 10^{-31})(1.67 \times 10^{-27})}{(5.29 \times 10^{-11})^2} = 3.63 \times 10^{-47}\,\text{N}.$$
- $F_e/F_g \approx 2.3 \times 10^{39}$. Inside atoms, gravity does not matter at all.
Answer: The electric force is about $2 \times 10^{39}$ times stronger
Common mistake
Thinking the larger charge feels the larger force. The formula has one number for the pair, $k|q_1q_2|/r^2$, and swapping the labels changes nothing. A $+8\,\mu\text{C}$ charge pushes a $+2\,\mu\text{C}$ charge exactly as hard as the $+2\,\mu\text{C}$ charge pushes back, as Newton's third law says. Check: the force on each charge is equal in size and opposite in direction.