Coulomb's law

Electromagnetism

Electrostatics
Electric charge · Coulomb's law · Electric field · Electric flux · Gauss's law · Electric potential · Electrostatic induction · Electric dipole moment · Polarization density

Magnetostatics
Ampère's law · Electric current · Magnetic field · Magnetization · Magnetic flux · Biot–Savart law · Magnetic dipole moment · Gauss's law for magnetism

Electrodynamics
Free space · Lorentz force law · emf · Electromagnetic induction · Faraday’s law · Lenz's law · Displacement current · Maxwell's equations · EM field · Electromagnetic radiation · Liénard–Wiechert potential · Maxwell tensor · Eddy current

Electrical Network
Electrical conduction · Electrical resistance · Capacitance · Inductance · Impedance · Resonant cavities · Waveguides

Covariant formulation
Electromagnetic tensor · EM Stress-energy tensor · Four-current · Electromagnetic four-potential

Scientists
Ampère · Coulomb · Faraday · Gauss · Heaviside · Henry · Hertz · Lorentz · Maxwell · Tesla · Volta · Weber · Ørsted

Coulomb's law is a law of physics describing the electrostatic interaction between electrically charged particles. It was studied and first published in 1783 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. Nevertheless, the dependence of the electric force with distance (inverse square law) had been proposed previously by Joseph Priestley^[1] and the dependence with both distance and charge had been discovered, but not published, by Henry Cavendish, prior to Coulomb's works.

Coulomb's law 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 of the charges and inversely proportional to the square of the distance between the two charges.

1 Scalar form
- 1.1 Electric field
2 Vector form
3 Electrostatic approximation
4 Table of derived quantities
5 See also
6 Notes
7 References
8 External links

Scalar form

Diagram describing the basic mechanism of Coulomb's law; like charges repel each other and opposite charges attract each other.

Coulomb's torsion balance

The scalar form of Coulomb's law will only describe the magnitude of the electrostatic force between two electric charges. If direction is required, then the vector form is required as well. The magnitude of the electrostatic force (F) on a charge (q₁) due to the presence of a second charge (q₂), is given by

$F = k_\mathrm{e} \frac{q_1q_2}{r^2}$

where r is the distance between the two charges and k_e a proportionality constant. A positive force implies a repulsive interaction, while a negative force implies an attractive interaction.^[2]

The proportionality constant k_e, called the Coulomb constant (sometimes called the Coulomb force constant), is related to defined properties of space and can be calculated exactly:^[3]

$\begin{align} k_\mathrm{e} &= \frac{1}{4 \pi \varepsilon_0} = \frac{c^2 \ \mu_0}{4 \pi} = c^2 \cdot 10^{-7} \ \mathrm{H} \cdot \mathrm{m}^{-1}\\ &= 8.987\ 551\ 787\ 368\ 176\ 4 \times 10^9 \ \mathrm{N \cdot m^2 \cdot C^{-2}}. \end{align}$

By definition in SI units, the speed of light in vacuum, denoted c,^[4] is 299,792,458 m·s⁻¹,^[5] and the magnetic constant (μ₀), is defined as 4π × 10⁻⁷ H·m⁻¹,^[6] leading to the consequential defined value for the electric constant (ε₀) as ε₀ = 1/(μ₀c²) ≈ 8.854187817×10⁻¹² F·m⁻¹.^[7] In cgs units, the unit charge, esu of charge or statcoulomb, is defined so that this Coulomb constant is 1 and dimensionless.

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 be equal to −2 with precision of at least 2.7±3.1×10⁻¹⁶.^[8]

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

Main article: Electric field

It follows from the Lorentz Force Law that the magnitude of the electric field (E) created by a single point charge (q) at a certain distance (r) is given by:

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

For a positive charge, the direction of the electric field points along lines directed radially away from the location of the point charge, while the direction is the opposite for a negative charge. The SI units of electric field are volts per metre or newtons per coulomb.

Vector form

In order to obtain both the magnitude and direction of the force on a charge, $q_1$ at position $\mathbf{r}_1$ , experiencing a field due to the presence of another charge, q₂ at position $\mathbf{r}_2$ , 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 $r$ is the separation of the two charges. This is simply the scalar definition of Coulomb's law with the direction given by the unit vector, $\mathbf{\hat{r}}_{21}$ , parallel with the line from charge $q_2$ to charge $q_1$ .^[9]

If both charges have the same sign (like charges) then the product $q_1q_2$ is positive and the direction of the force on $q_1$ is given by $\mathbf{\hat{r}}_{21}$ ; the charges repel each other. If the charges have opposite signs then the product $q_1q_2$ is negative and the direction of the force on $q_1$ is given by $-\mathbf{\hat{r}}_{21}$ ; 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, $q$ , due to a system of $N$ 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 $q_i$ and $\mathbf{r}_i$ are the magnitude and position respectively of the $i^{th}$ charge, $\mathbf{\hat{R}}_{i}$ is a unit vector in the direction of $\mathbf{R}_{i} = \mathbf{r} - \mathbf{r}_i$ (a vector pointing from charge $q_i$ to charge $q$ ), and $R_{i}$ is the magnitude of $\mathbf{R}_{i}$ (the separation between charges $q_i$ and $q$ ).^[9]

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 $dq$ .

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

$dq = \lambda(\mathbf{r^\prime})dl^\prime$ .^[10]

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

$dq = \rho(\mathbf{r^\prime})\,dV^\prime.$ ^[9]

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

$\mathbf{F} = {q^\prime \over 4\pi\varepsilon_0}\int dq {\mathbf{r} - \mathbf{r^\prime} \over |\mathbf{r} - \mathbf{r^\prime}|^3}.$

Graphical representation

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

A graphical representation of Coulomb's law.

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}_{21}= {1 \over 4\pi\varepsilon_0}{q_1 q_2 \over r^2}\mathbf{\hat{r}}_{21} \$

$\mathbf{F}_{21}= q_1 \mathbf{E}_{21}$

Electric field (at 1 by 2)

$\mathbf{E}_{21}= {1 \over 4\pi\varepsilon_0}{q_2 \over r^2}\mathbf{\hat{r}}_{21} \$

Relationship

$\mathbf{F}_{21}=-\mathbf{\nabla}U_{21}$

$\mathbf{E}_{21}=-\mathbf{\nabla}V_{21}$

Scalar quantity

Potential energy (at 1 by 2)

$U_{21}={1 \over 4\pi\varepsilon_0}{q_1 q_2 \over r} \$

$U_{21}=q_1 V_{21} \$

Potential (at 1 by 2)

$V_{21}={1 \over 4\pi\varepsilon_0}{q_2 \over r}$

Notes

↑ Robert S. Elliott (1999). Electromagnetics: History, Theory, and Applications. ISBN 978-0-7803-5384-8. http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0780353846.html
↑ Coulomb's law, Hyperphysics
↑ Coulomb's constant, Hyperphysics
↑ 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 and continues to be commonly used. See NIST Special Publication 330, Appendix 2, p. 45
↑ [1]
↑ [2]
↑ http://physics.nist.gov/cgi-bin/cuu/Value?ep0
↑ Williams, Faller, Hill, E.; Faller, J.; Hill, H. (1971). "New Experimental Test of Coulomb's Law: A Laboratory Upper Limit on the Photon Rest Mass". Physical Review Letters 26: 721–724. doi:10.1103/PhysRevLett.26.721. http://prola.aps.org/abstract/PRL/v26/i12/p721_1
↑ ^9.0 ^9.1 ^9.2 Coulomb's law, University of Texas
↑ 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.

External links

Coulomb's Law on Project PHYSNET.
Electricity and the Atom — a chapter from an online textbook
A maze game for teaching Coulomb's Law—a game created by the Molecular Workbench software
The inverse cube law The inverse cube law for dipoles (PDF file) by Eng. Xavier Borg
Electric Charges, Polarization, Electric Force, Coulomb's Law Walter Lewin, 8.02 Electricity and Magnetism, Spring 2002: Lecture 1 (video). MIT OpenCourseWare. License: Creative Commons Attribution-Noncommercial-Share Alike.