# Coulomb's law

## Overview

Template:Electromagnetism Coulomb's law, developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form as follows:

The magnitude of the electrostatic force between two point electric charges is directly proportional to the product of the magnitudes of each charge and inversely proportional to the square of the distance between the charges.

## Scalar form

If one does not require the specific direction of the force then the simplified, scalar, version of Coulomb's law will suffice. The magnitude of the force on a charge, $\scriptstyle{q_1}$, due to the presence of a second charge, $\scriptstyle{q_2}$, is given by the magnitude of

$F = {1 \over 4\pi\varepsilon_0}\frac{q_1q_2}{r^2}$,

where $\scriptstyle{r}$ is the separation of the charges and $\scriptstyle{\varepsilon_0}$ is the electric constant. A positive force implies a repulsive interaction, while a negative force implies an attractive interaction.[1]

The prefactor, termed the Coulomb's constant ($\scriptstyle{k_e}$), is:

\begin{align} k_e &= \frac{1}{4\pi\varepsilon_0} = \frac{\mu_0\ {c_0}^2}{4 \pi} = 10^{-7}\ {c_0}^2 \\ &= 8.987\ 551\ 787\ \times 10^9 \\ \end{align}

$\approx 9 \times 10^9$   Nm2C−2 (also mF−1).[2]

In SI units the speed of light in vacuum c0 is defined[3] as the numerical value c0 = 299 792 458 m s−1 (See c0) and the magnetic constant μ0 is defined as 4π x 10−7 H · m−1 (See μ0), leading to the definition for the electric constant of ε0 = 1/(μ0c02) ≈ 8.854 187 817 x 10−12 F m−1 (See NIST ε0). In cgs units, the unit charge, esu of charge or statcoulomb, is defined so that this Coulomb force constant is 1.

This formula says that the magnitude of the force is directly proportional to the magnitude of the charges of each object and inversely proportional to the square of the distance between them. The exponent in Coulomb's Law has been found to differ from −2 by less than one in a billion.[4]

When measured in units that people commonly use (such as SI—see International System of Units), the electrostatic force constant, $\scriptstyle{k_e}$, is numerically much much larger than the universal gravitational constant $\scriptstyle{G}$.[5] This means that for objects with charge that is of the order of a unit charge (C) and mass of the order of a unit mass (kg), the electrostatic forces will be so much larger than the gravitational forces that the latter force can be ignored. This is not the case when Planck units are used and both charge and mass are of the order of the unit charge and unit mass. However, charged elementary particles have mass that is far less than the Planck mass while their charge is about the Planck charge so that, again, gravitational forces can be ignored. For example, the electrostatic force between an electron and a proton, which constitute a hydrogen atom, is almost 40 orders of magnitude greater than the gravitational force between them.[6]

Coulomb's law can also be interpreted in terms of atomic units with the force expressed in Hartrees per Bohr radius, the charge in terms of the elementary charge, and the distances in terms of the Bohr radius.

### Electric field

It follows from the Lorentz Force Law that the magnitude of the electric field $\scriptstyle{\mathbf{E}}$ created by a single point charge $\scriptstyle{q}$ is given by

$E = {1 \over 4\pi\varepsilon_0}\frac{q}{r^2}$

For a positive charge $\scriptstyle{q}$, the direction of $\scriptstyle{\mathbf{E}}$ points along lines directed radially away from the location of the point charge, while the direction is the opposite for a negative charge. The units of electric field are volts per meter or newtons per coulomb.

## Vector form

In order to obtain both the magnitude and direction of the force on a charge, $\scriptstyle{q_1}$ at position $\scriptstyle{\mathbf{r}_1}$, experiencing a field due to the presence of another charge, $\scriptstyle{q_2}$ at position $\scriptstyle{\mathbf{r}_2}$, the full vector form of Coulomb's law is required.

$\mathbf{F} = {1 \over 4\pi\varepsilon_0}{q_1q_2(\mathbf{r}_1 - \mathbf{r}_2) \over |\mathbf{r}_1 - \mathbf{r}_2|^3} = {1 \over 4\pi\varepsilon_0}{q_1q_2 \over r^2}\mathbf{\hat{r}}_{21}$,

where $\scriptstyle{r}$ is the separation of the two charges. Note that this is simply the scalar definition of Coulomb's law with the direction given by the unit vector, $\scriptstyle{\mathbf{\hat{r}}_{21}}$, parallel with the line from charge $\scriptstyle{q_2}$ to charge $\scriptstyle{q_1}$.[6]

If both charges have the same sign (like charges) then the product $\scriptstyle{q_1q_2}$ is positive and the direction of the force on $\scriptstyle{q_1}$ is given by $\scriptstyle{\mathbf{\hat{r}}_{21}}$; the charges repel each other. If the charges have opposite signs then the product $\scriptstyle{q_1q_2}$ is negative and the direction of the force on $\scriptstyle{q_1}$ is given by $-\scriptstyle{\mathbf{\hat{r}}_{21}}$; the charges attract each other.

### System of discrete charges

The principle of linear superposition may be used to calculate the force on a small test charge, $\scriptstyle{q}$, due to a system of $\scriptstyle{N}$ discrete charges:

$\mathbf{F}(\mathbf{r}) = {q \over 4\pi\varepsilon_0}\sum_{i=1}^N {q_i(\mathbf{r} - \mathbf{r}_i) \over |\mathbf{r} - \mathbf{r}_i|^3} = {q \over 4\pi\varepsilon_0}\sum_{i=1}^N {q_i \over R_{i}^2}\mathbf{\hat{R}}_{i}$,

where $\scriptstyle{q_i}$ and $\scriptstyle{\mathbf{r}_i}$ are the magnitude and position respectively of the $\scriptstyle{i^{th}}$ charge, $\scriptstyle{\mathbf{\hat{R}}_{i}}$ is a unit vector in the direction of $\scriptstyle{\mathbf{R}_{i} = \mathbf{r} - \mathbf{r}_i}$ (a vector pointing from charge $\scriptstyle{q_i}$ to charge $\scriptstyle{q}$), and $\scriptstyle{R_{i}}$ is the magnitude of $\scriptstyle{\mathbf{R}_{i}}$ (the separation between charges $\scriptstyle{q_i}$ and $\scriptstyle{q}$).[6]

### Continuous charge distribution

For a charge distribution an integral over the region containing the charge is equivalent to an infinite summation, treating each infinitesimal element of space as a point charge $\scriptstyle{dq}$.

For a linear charge distribution (a good approximation for charge in a wire) where $\scriptstyle{\lambda(\mathbf{r^\prime})}$ gives the charge per unit length at position $\scriptstyle{\mathbf{r^\prime}}$, and $\scriptstyle{dl^\prime}$ is an infinitesimal element of length,

$dq = \lambda(\mathbf{r^\prime})dl^\prime$.[7]

For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where $\scriptstyle{\sigma(\mathbf{r^\prime})}$ gives the charge per unit area at position $\scriptstyle{\mathbf{r^\prime}}$, and $\scriptstyle{dA^\prime}$ is an infinitesimal element of area,

$dq = \sigma(\mathbf{r^\prime})dA^\prime$.

For a volume charge distribution (such as charge within a bulk metal) where $\scriptstyle{\rho(\mathbf{r^\prime})}$ gives the charge per unit volume at position $\scriptstyle{\mathbf{r^\prime}}$, and $\scriptstyle{dV^\prime}$ is an infinitesimal element of volume,

$dq = \rho(\mathbf{r^\prime})dV^\prime$.[6]

The force on a small test charge $\scriptstyle{q^\prime}$ at position $\scriptstyle{\mathbf{r}}$ is given by

$\mathbf{F} = q^\prime\int dq {\mathbf{r} - \mathbf{r^\prime} \over |\mathbf{r} - \mathbf{r^\prime}|^3}$.

### Graphical representation

Below is a graphical representation of Coulomb's law, when $\scriptstyle{q_1q_2 > 0}$. The vector $\scriptstyle{\mathbf{F}_1}$ is the force experienced by $\scriptstyle{q_1}$. The vector $\scriptstyle{\mathbf{F}_2}$ is the force experienced by $\scriptstyle{q_2}$. Their magnitudes will always be equal. The vector $\scriptstyle{\mathbf{r}_{21}}$ is the displacement vector between two charges ($\scriptstyle{q_1}$ and $\scriptstyle{q_2}$).

## Electrostatic approximation

In either formulation, Coulomb's law is fully accurate only when the objects are stationary, and remains approximately correct only for slow movement. These conditions are collectively known as the electrostatic approximation. When movement takes place, magnetic fields are produced which alter the force on the two objects. The magnetic interaction between moving charges may be thought of as a manifestation of the force from the electrostatic field but with Einstein's theory of relativity taken into consideration.

## Table of derived quantities

Particle property Relationship Field property
Vector quantity
 Force (on 1 by 2) $\mathbf{F}_{12}= {1 \over 4\pi\varepsilon_0}{q_1 q_2 \over r^2}\mathbf{\hat{r}}_{21} \$
$\mathbf{F}_{12}= q_1 \mathbf{E}_{12}$
 Electric field (at 1 by 2) $\mathbf{E}_{12}= {1 \over 4\pi\varepsilon_0}{q_2 \over r^2}\mathbf{\hat{r}}_{21} \$
Relationship $\mathbf{F}_{12}=-\mathbf{\nabla}U_{12}$ $\mathbf{E}_{12}=-\mathbf{\nabla}V_{12}$
Scalar quantity
 Potential energy (at 1 by 2) $U_{12}={1 \over 4\pi\varepsilon_0}{q_1 q_2 \over r} \$
$U_{12}=q_1 V_{12} \$
 Potential (at 1 by 2) $V_{12}={1 \over 4\pi\varepsilon_0}{q_2 \over r}$

## Footnotes

1. Coulomb's law, Hyperphysics
2. Coulomb's constant, Hyperphysics
3. Current practice is to use c0 to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose. See NIST Special Publication 330, Appendix 2, p. 45
4. Williams, Faller, Hill (1971), "New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass", Physical Review Letters, 26: 721–724
5. This large ratio has led to the Dirac large numbers hypothesis.
6. Coulomb's law, University of Texas
7. Charged rods, PhysicsLab.org

## References

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.