Coulomb's law

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Coulomb's law or Coulomb's inverse-square law is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. Coulomb's law has been tested heavily and all observations are consistent with the law.

History

Charles Augustin de Coulomb

Early investigators who suspected that the electrical force diminished with distance as the gravitational force did (i.e., as the inverse square of the distance) included Daniel Bernoulli and Alessandro Volta, both of whom measured the force between plates of a capacitor, and Aepinus who supposed the inverse-square law in 1758.

Based on experiments with charged spheres, Joseph Priestley of England was among the first to propose that electrical force followed an inverse-square law, similar to Newton's law of universal gravitation. However, he did not generalize or elaborate on this. In 1767, he conjectured that the force between charges varied as the inverse square of the distance, saying:

Coulomb’s torsion balance

May we not infer from this experiment, that the attraction of electricity is subject to the same laws with that of gravitation, and is therefore according to the squares of the distances; since it is easily demonstrated that, were the earth in the form of a shell, a body in the inside of it would not be attracted to one side more than another?

In 1769, Scottish physicist John Robison announced that, according to his measurements, the force of repulsion between two spheres with charges of the same sign varied as x^-2.06.

The dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, in the early 1770s by Henry Cavendish of England.

Finally, in 1785, the French physicist Charles Augustin de Coulomb published his first three reports of electricity and magnetism where he stated his law. This publication was essential to the development of the theory of electromagnetism. He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The law

Coulomb's law states that the magnitude of the Electrostatics force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distances between them.

If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.

The scalar and vector forms of the mathematical equation are

$|\boldsymbol{F}|=k_e{|q_1q_2|\over r^2}$

and

$\boldsymbol{F}=k_e{q_1q_2\boldsymbol{\hat{r}_{21}}\over r_{21}^2}$

,   respectively (here, $\boldsymbol{\hat{r}_{21}} = \frac{\boldsymbol{r_{21}}}{|\mathbf{r_{21}}|}$).

Units

Electromagnetic theory is usually expressed using the standard SI units for the force F, quantity of charge q, and radial distance r, and where k e  = 1 ⁄ 4πε 0 ε . Where ε 0  is the permittivity of free space and ε  is the relative permittivity of the material in which the charges are immersed. Coulomb's law and Coulomb's constant can also be interpreted in terms of atomic units, or electrostatic units or Gaussian units. In atomic units the force is expressed in Hartrees per Bohr radius, the charge in terms of the elementary charge, and the distances in terms of the Bohr radius. In electrostatic units and Gaussian units, the unit charge (esu or statcoulomb) is defined in such a way that the Coulomb constant k disappears because it has the value of one and becomes dimensionless. The standard SI units will be used below.

The SI derived units for the electric field are volts per meter, newtons per coulomb, and teslas meters per second.

An electric field

If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.

The magnitude of the electric field force, E in vacuum, is invertible from Coulomb's law. Since E = F ⁄ Q it follows from the Coulomb's law that the magnitude of the electric field E created by a single point charge q at a certain distance r in vacuum is given by:

$|\boldsymbol{E}|={1\over4\pi\varepsilon_0}{|q|\over r^2}$.

An electric field is a vector field which associates to each point of the space the Coulomb force that will experience a test unity charge. Given the electric field, the strength and direction of a force F on a quantity charge q in an electric field E is determined by the electric field. 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 towards for a negative charge.

Coulomb's constant

Main article: Coulomb's constant

Coulomb's constant (denoted k e ) is a proportionality factor also called the electric force constant or electrostatic constant, hence the subscript e, that appears in Coulomb's law as well as in other electric-related formulas.

The exact value of Coulomb's constant k e  comes from three of the fundamental, invariant quantities that define free space in the SI system: the speed of light c 0 , magnetic permeability μ 0 , and electric permittivity ε 0 , related by Maxwell as

$\frac{1}{\mu_0\varepsilon_0}=c_0^2.$

Because of the way the SI base unit system made the natural units for electromagnetism, the speed of light in vacuum c 0  is 299,792,458 m s⁻¹, the magnetic permeability μ 0  of free space is 4π·10⁻⁷H m⁻¹, and the electric permittivity of free space is ε 0  = 1 ⁄ (μ 0 c 2
0 ) ≈ 8.85418782×10⁻¹² F·m⁻¹, so that

$\begin{align} k_e &= \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}=c_0^2\cdot10^{-7}\mathrm{H\ m}^{-1}\\ &= 8.987\ 551\ 787\ 368\ 176\ 4\cdot10^9\mathrm{N\ m^2\ C}^{-2}. \end{align}$

Conditions for validity

There are two conditions to be fulfilled for the validity of Coulomb’s law:

1. The charges considered must be point charges.
2. They should be stationary with respect to each other.

Scalar form

The absolute value of the force F between two point charges q and Q relates to the distance between the point charges and to the simple product of their charges by Coulomb's law. The diagram shows that like charges repel each other, and opposite charges attract each other.

The scalar form of Coulomb's Law relates the magnitude and sign of the electrostatic force F, acting simultaneously on two point charges q 1  and q 2 :

$|\boldsymbol{F}|=k_e{|q_1q_2|\over r^2}$

where r is the separation distance and k e  is Coulomb's constant. If the product q 1 q 2 is positive, the force between them is repulsive; if q 1 q 2 is negative, the force between them is attractive.

The law of superposition allows this law to be extended to include any number of point charges, to derive the force on any one point charge by a vector addition of these individual forces acting alone on that point charge. The resulting force vector happens to be parallel to the electric field vector at that point, with that point charge removed.

Vector form

In the image, the vector F 1  is the force experienced by q 1 , and the vector F 2  is the force experienced by q 2 . When q 1 q 2  > 0 the forces are repulsive (as in the image) and when q 1 q 2  < 0 the forces are attractive (opposite to the image). Their magnitudes will always be equal.

Coulomb's law states that the force $\boldsymbol{F}$ on a charge, $q_1$ at position $\boldsymbol{r_1}$, experiencing an electric field due to the presence of another charge, $q_2$ at position $\boldsymbol{r_2}$ in vacuum is:

$\boldsymbol{F}={q_1q_2\over4\pi\varepsilon_0}{(\boldsymbol{r_1-r_2})\over|\boldsymbol{r_1-r_2}|^3}={q_1q_2\over4\pi\varepsilon_0}{\boldsymbol{\hat{r}_{21}}\over r_{21}^2},$

where $\boldsymbol{r_{21}}=\boldsymbol{r_1-r_2}$ and $\varepsilon_0$ is the electric constant. This is simply the scalar definition of Coulomb's law with the direction given by the unit vector, $\boldsymbol{\hat{r}_{21}}$, parallel with the line from charge $q_2$ to charge $q_1$.

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 $\boldsymbol{\hat{r}_{21}}$; 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 $-\boldsymbol{\hat{r}_{21}}$; 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, $q$ at position $\boldsymbol{r}$, due to a system of $N$ discrete charges in vacuum:

$\boldsymbol{F(r)}={q\over4\pi\varepsilon_0}\sum_{i=1}^Nq_i{\boldsymbol{r-r_i}\over|\boldsymbol{r-r_i}|^3}={q\over4\pi\varepsilon_0}\sum_{i=1}^Nq_i{\boldsymbol{\widehat{R_i}}\over|\boldsymbol{R_i}|^2},$

where $q_i$ and $\boldsymbol{r_i}$ are the magnitude and position respectively of the $i^{th}$ charge, $\boldsymbol{\widehat{R_i}}$ is a unit vector in the direction of $\boldsymbol{R}_{i} = \boldsymbol{r} - \boldsymbol{r}_i$ (a vector pointing from charges $q_i$ to $q$).

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 $\lambda(\boldsymbol{r'})$ gives the charge per unit length at position $\boldsymbol{r'}$, and $dl'$ is an infinitesimal element of length,

$dq = \lambda(\boldsymbol{r'})dl'$.

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

$dq = \sigma(\boldsymbol{r'})\,dA'.$

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

$dq = \rho(\boldsymbol{r'})\,dV'.$

The force on a small test charge $q'$ at position $\boldsymbol{r}$ in vacuum is given by

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

Experimental verification of Coulomb's law

Experiment to verify Coulomb's law.

It is possible to verify Coulomb's law with a simple experiment. Let's consider two small spheres of mass m and same-sign charge q, hanging from two ropes of negligible mass and length l. The forces acting on each sphere are three: the weight mg, the rope tension T and the electric force F.

In the equilibrium state:

$T \ \sin \theta_1 =F_1 \,\!$

(1)

and:

$T \ \cos \theta_1 =mg \,\!$

(2)

Dividing (1) over (2):

$\frac {\sin \theta_1}{\cos \theta_1 }= \frac {F_1}{mg}\Rightarrow F_1= mg \tan \theta_1$

(3)

Being $L_1 \,\!$ the distance between the charged spheres; the repulsion force between them $F_1 \,\!$, assuming Coulomb's law is correct, is equal to

$F_1 = \frac{q^2}{4 \pi \epsilon_0 L_1^2}$

(Coulomb's law)

so:

$\frac{q^2}{4 \pi \epsilon_0 L_1^2}=mg \tan \theta_1 \,\!$

(4)

If we now decharge one of the spheres, and we put it in contact with the charged sphere, each one of them adquiers a charge q/2. In the equilibrium state, the distance between the charges will be $L_2<L_1 \,\!$ and the repulsion force between them will be:

$F_2 = \frac{{(q/2)}^2}{4 \pi \epsilon_0 L_2^2}=\frac{q^2/4}{4 \pi \epsilon_0 L_2^2} \,\!$

(5)

We know that $F_2= mg. \tan \theta_2 \,\!$. And:

$\frac{\frac{q^2}{4}}{4 \pi \epsilon_0 L_2^2}=mg. \tan \theta_2$

Dividing (3) over (4), we get:

$\frac{\left( \cfrac{q^2}{4 \pi \epsilon_0 L_1^2} \right)}{\left(\cfrac{q^2/4}{4 \pi \epsilon_0 L_2^2}\right)}= \frac{mg \tan \theta_1}{mg \tan \theta_2} \Longrightarrow 4 {\left ( \frac {L_2}{L_1} \right ) }^2= \frac{ \tan \theta_1}{ \tan \theta_2}$

(6)

Measuring the angles $\theta_1 \,\!$ and $\theta_2 \,\!$ and the distance bethween the charges $L_1 \,\!$ and $L_2 \,\!$ is possible to verify that the equality is true, taking into account the experimental error. In practice, angles can be difficult to measure, so if the length of the ropes is sufficiently big, the angles will be enough small to make the following aproximation:

$\tan \theta \approx \sin \theta= \frac{\frac{L}{2}}{l}=\frac{L}{2l}\Longrightarrow\frac{ \tan \theta_1}{ \tan \theta_2}\approx \frac{\frac{L_1}{2l}}{\frac{L_2}{2l}}$

(7)

Using this approximation, the relationship (6) turns into this much more simple expression:

$\frac{\frac{L_1}{2l}}{\frac{L_2}{2l}}\approx 4 {\left ( \frac {L_2}{L_1} \right ) }^2 \Longrightarrow \,\!$ $\frac{L_1}{L_2}\approx 4 {\left ( \frac {L_2}{L_1} \right ) }^2\Longrightarrow \frac{L_1}{L_2}\approx\sqrt[3]{4} \,\!$

(8)

In this way, the verification is limited to measuring the distance between the charges and check that the division approximates the theoretical value.

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 that alter the force on the two objects are produced. 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. Other theories like Weber electrodynamics predict other velocity-dependent corrections to Coulomb's law.

Atomic forces

Coulomb's law holds even within the atoms, correctly describing the force between the positively charged nucleus and each of the negatively charged electrons. This simple law also correctly accounts for the forces that bind atoms together to form molecules and for the forces that bind atoms and molecules together to form solids and liquids.

Generally, as the distance between ions increases, the energy of attraction approaches zero and ionic bonding is less favorable. As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable.

Table of derived quantities (relevant for vacuum only)

At/on 1 by 2	Particle property	Relationship	Field property
Vector quantity	Force $\boldsymbol{F_{21}}={q_1q_2\over4\pi\varepsilon_0}{\boldsymbol{\widehat{R_{21}}}\over|\boldsymbol{R_{21}}|^2}$	$\boldsymbol{F_{21}}=q_1\boldsymbol{E_{21}}$	Electric field $\boldsymbol{E_{21}}={q_2\over4\pi\varepsilon_0}{\boldsymbol{\widehat{R_{21}}}\over|\boldsymbol{R_{21}}|^2}$
Relationship	$\boldsymbol{F_{21}}=-\nabla U_{21}$		$\boldsymbol{E_{21}}=-\nabla V_{21}$
Scalar quantity	Electric energy $U_{21}={1\over4\pi\varepsilon_0}{q_1q_2\over|\boldsymbol{R_{21}}|}$	$U_{21}=q_1V_{21}$	Electric potential $V_{21}={1\over4\pi\varepsilon_0}{q_2\over|\boldsymbol{R_{21}}|}$

See also

• Biot–Savart law
• Method of image charges
• Electromagnetic force
• Molecular modelling
• Static forces and virtual-particle exchange
• Darwin Lagrangian
• Newton's Law of Universal Gravitation, which uses a similar structure, but for mass instead of charge.

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
• 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.