# Coulomb's Law of Electrostatics

## Physical Law

Let $a$ and $b$ be stationary particles in a vacuum, each carrying an electric charge of $q_a$ and $q_b$ respectively.

Then $a$ and $b$ exert a force upon each other whose magnitude and direction are given by **Coulomb's law (of electrostatics)**:

- $\mathbf F_{a b} \propto \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$

where:

- $\mathbf F_{a b}$ is the force exerted on $b$ by the electric charge on $a$
- $\mathbf r_{a b}$ is the displacement vector from $a$ to $b$
- $r$ is the distance between $a$ and $b$.
- the constant of proportion is defined as being positive.

By exchanging $a$ and $b$ in the above, it is seen that $b$ exerts the same force on $a$ as $a$ does on $b$, but in the opposite direction.

## SI Units

In SI units, the law becomes:

- $\mathbf F_{a b} = \dfrac 1 {4 \pi \varepsilon_0} \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$

where:

- $q_a$ and $q_b$ are measured in coulombs
- $r$ is measured in metres
- $F_{a b}$ is measured in newtons
- $\varepsilon_0$ denotes the vacuum permittivity:
- $\varepsilon_0 \approx 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm C^2 \, \mathrm N^{-1} \, \mathrm m^{-2}$

Thus the equation becomes:

- $\mathbf F_{a b} = \dfrac 1 {4 \pi \varepsilon_0} \dfrac {q_a q_b {\mathbf r_{a b} } } {r^3}$

## Also presented as

- $\mathbf F_{a b} \propto \dfrac {q_a q_b \hat {\mathbf r}_{a b} } {r^2}$

where $\hat {\mathbf r}_{a b}$ is the unit vector in the direction from $a$ to $b$.

## Also known as

**Coulomb's Law of Electrostatics** is also known as just **Coulomb's Law**.

## Source of Name

This entry was named for Charles-Augustin de Coulomb.

## Historical Note

Charles-Augustin de Coulomb proposed what is now referred to as Coulomb's Law in the year $1785$.

## Sources

- 1951: B. Hague:
*An Introduction to Vector Analysis*(5th ed.) ... (previous) ... (next): Chapter $\text {V}$: Further Applications of the Operator $\nabla$: $9$. The Vector Field $\map \grad {k / r}$ - 1990: I.S. Grant and W.R. Phillips:
*Electromagnetism*(2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics - 1990: I.S. Grant and W.R. Phillips:
*Electromagnetism*(2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.1$ Electric Charge