# Coulomb's Law

## Table of Contents

## Scalar Form #

For any two point charges, Coulomb’s Law says that the electrostatic force between them is given by:

$$ F = \frac{k q_1 q_2}{r^2} $$

Where

- \( \small k \) is the electrostatic constant equal to \( \frac{1}{4 \pi \epsilon_0} \), with \( \small \epsilon_0 \) being the permittivity of free space,
- \( \small q_1 \) and \( \small q_2 \) are the magnitudes of the electric charges of the two point charges, and
- \( \small r \) is the distance between the two point charges.

If the two point charges have the same sign, the electrostatic force is repulsive. If the two point charges have opposite signs, the electrostatic force is attractive. The force experienced by each point charge is equal in magnitude and opposite in direction.

## Vector Form #

Coulomb’s Law can also be expressed in vector form, which specifies direction directly rather than through intuition. For two point charges \( q_1 \) and \( q_2 \), the electrostatic force on charge \( q_1 \) is given by:

$$ \vec{F} = k q_1 q_2 \frac{\bold{r_1}-\bold{r_2}}{|\bold{r_1} - \bold{r_2}|^3} = k q_1 q_2 \frac{\mathbf{\hat{r}_{12}}}{{|\bold{r_1} - \bold{r_2}|^2}} $$

Where

- \( \small k \) is the electrostatic constant equal to \( \frac{1}{4 \pi \epsilon_0} \), with \( \small epsilon_0 \) being the permittivity of free space,
- \( \small q_1 \) and \( \small q_2 \) are the magnitudes of the electric charges of the two point charges,
- \( \small \bold{r_1} \) and \( \small \bold{r_2} \) are the position vectors of the two point charges,
- \( \small \bold{r_1} - \bold{r_2} \) is the vector pointing from \( \bold{r_2} \) to \( \bold{r_1} \),
- \( \small | . | \) denotes the magnitude of a vector, and
- \( \small \mathbf{\hat{r}_{12}} \) is the unit vector pointing from \( \small \bold{r_2} \) to \( \small \bold{r_1} \)

For multiple point charges, the total force on a point charge is the vector sum of the forces on that point charge from each other point charge.