Coulomb’s Law

Coulomb’s law governs how electric charges interact with each other when they are both at rest.

Empirical Law: Coulomb's Law

Every pair of point charges $q_1$ and $q_2$ which are both at rest interact with a pair of forces, ${\mathbf{F}}_{\text{on }{q}_1}$ and ${\mathbf{F}}_{\text{on }{q}_2}$, which we call electrostatic forces. We refer to which depend on the charges $q_1$ and $q_2$, the distance $r$ between them and the permittivity $\epsilon$ of their environment.

• Magnitude - the electrostatic forces ${\mathbf{F}}_{\text{on }{q}_1}$ and ${\mathbf{F}}_{\text{on }{q}_2}$ have the same magnitude which depends on $q_1$ and $q_2$, the square of the distance $r$ between them and is proportional to a constant $\epsilon$ which is a characteristic of the environment of the two charges called the permittivity:

$$||{\mathbf{F}}_{\text{on }{q}_1}||=||{\mathbf{F}}_{\text{on }{q}_2}||=\frac{1}{4\pi \epsilon }\frac{|q_1{q}_2|}{r^2}$$

• Direction - the electrostatic forces always act along the line joining $q_1$ and $q_2$. It is repulsive if both charges have the same algebraic sign, and it is attractive if they have different signs.

Equations for the electrostatic forces as vectors can be written using the normalized difference between their positions, where ${\hat{\mathbf{r}}}_{q_1\to q_2}=\frac{1}{||{\mathbf{r}}_{q_2}-{\mathbf{r}}_{q_1}||}{\mathbf{r}}_{q_2}-{\mathbf{r}}_{q_1}$ and ${\hat{\mathbf{r}}}_{q_2\to q_1}=\frac{1}{||{\mathbf{r}}_{q_1}-{\mathbf{r}}_{q_2}||}{\mathbf{r}}_{q_1}-{\mathbf{r}}_{q_2}$:

$${\mathbf{F}}_{\text{on }{q}_2}=\frac{1}{4\pi \epsilon }\frac{q_1{q}_2}{r^2}{\hat{\mathbf{r}}}_{q_1\to q_2}$$ $${\mathbf{F}}_{\text{on }{q}_1}=\frac{1}{4\pi \epsilon }\frac{q_1{q}_2}{r^2}{\hat{\mathbf{r}}}_{q_2\to q_1}$$