Coulomb's Law

Charles Augustin de Coulomb measured, using a sensitive torsion balance, exactly how the force between two small charged spheres depends on their charge and their separation. His result, Coulomb's law, is the single most important quantitative law in electrostatics:

The force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of the two charges, and inversely proportional to the square of the distance between them; it acts along the straight line joining the two charges.

For two point charges $q_1$ and $q_2$ separated by distance $r$, the magnitude of the force is

where $\epsilon_0 = 8.854\times10^{-12}\ \text{C}^2\text{N}^{-1}\text{m}^{-2}$ is the permittivity of free space, and the combination $k=1/4\pi\epsilon_0 \approx 9\times10^9\ \text{N}\,\text{m}^2/\text{C}^2$ is the Coulomb constant. (Writing the constant as $1/4\pi\epsilon_0$ rather than simply as $k$ looks awkward here, but it is chosen deliberately so that the $4\pi$ cancels out neatly later, when Gauss's theorem is written for a sphere.) When the two charges sit in some other medium instead of vacuum, $\epsilon_0$ is replaced by $\epsilon=\epsilon_r\epsilon_0$, where $\epsilon_r$ is the medium's relative permittivity (dielectric constant); the force is always weaker in a medium than in vacuum, since $\epsilon_r>1$ for every real material.

Written in full vector form, the force $\vec{F}_{21}$ exerted BY charge $q_1$ ON charge $q_2$, with $\hat{r}_{21}$ the unit vector pointing from $q_1$ to $q_2$, is

The sign of the product $q_1q_2$ automatically fixes the direction: if $q_1$ and $q_2$ have the same sign, $q_1q_2>0$ and $\vec{F}_{21}$ points along $+\hat{r}_{21}$, i.e. away from $q_1$ -- a repulsive force; if they have opposite signs, $q_1q_2<0$ and $\vec{F}_{21}$ points along $-\hat{r}_{21}$, i.e. toward $q_1$ -- an attractive force. By Newton's third law, the force $\vec{F}_{12}$ that $q_2$ exerts back on $q_1$ is exactly equal in magnitude and opposite in direction: $\vec{F}_{12} = -\vec{F}_{21}$. …