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Coulomb's law

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Fundamental physical law of electromagnetism

The magnitude of the electrostaticforceF between twopoint chargesq₁ andq₂ is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Like charges repel each other, and opposite charges attract each other.

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Coulomb's inverse-square law, or simplyCoulomb's law, is an experimentallaw^[1] ofphysics that calculates the amount of force between twoelectrically charged particles at rest. This electric force is conventionally called theelectrostatic force orCoulomb force.^[2] Although the law was known earlier, it was first published in 1785 by French physicistCharles-Augustin de Coulomb. Coulomb's law was essential to the development of thetheory of electromagnetism and maybe even its starting point,^[1] as it allowed meaningful discussions of the amount of electric charge in a particle.^[3]

The law states that the magnitude, or absolute value, of the attractive or repulsive electrostaticforce between two pointcharges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the square of the distance between them.^[4] Coulomb discovered that bodies with like electrical charges repel:

It follows therefore from these three tests, that the repulsive force that the two balls – [that were] electrified with the same kind of electricity – exert on each other, follows the inverse proportion of the square of the distance.^[5]

Coulomb also showed that oppositely charged bodies attract according to an inverse-square law: $|F|=k_{\text{e}}{\frac {|q_{1}||q_{2}|}{r^{2}}}$

Here,k_e is a constant,q₁ andq₂ are the quantities of each charge, and the scalarr is the distance between the charges.

The force is along the straight line joining the two charges. If the charges have the same sign, the electrostatic force between them makes them repel; if they have different signs, the force between them makes them attract.

Being aninverse-square law, the law is similar toIsaac Newton's inverse-squarelaw of universal gravitation, but gravitational forces always make things attract, while electrostatic forces make charges attract or repel. Also, gravitational forces are much weaker than electrostatic forces.^[2] Coulomb's law can be used to deriveGauss's law, and vice versa. In the case of a single point charge at rest, the two laws are equivalent, expressing the same physical law in different ways.^[6] The law has beentested extensively, and observations have upheld the law on the scale from 10⁻¹⁶ m to 10⁸ m.^[6]

History

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Ancient cultures around theMediterranean knew that certain objects, such as rods ofamber, could be rubbed with cat's fur to attract light objects like feathers and pieces of paper.Thales of Miletus made the first recorded description ofstatic electricity around 600 BC,^[7] when he noticed thatfriction could make a piece of amber attract small objects.^[8]^[9]

In 1600, English scientistWilliam Gilbert made a careful study of electricity and magnetism, distinguishing thelodestone effect from static electricity produced by rubbing amber.^[8] He coined theNeo-Latin wordelectricus ("of amber" or "like amber", fromἤλεκτρον [elektron], the Greek word for "amber") to refer to the property of attracting small objects after being rubbed.^[10] This association gave rise to the English words "electric" and "electricity", which made their first appearance in print inThomas Browne'sPseudodoxia Epidemica of 1646.^[11]

Early investigators of the 18th century who suspected that the electricalforce diminished with distance as the force ofgravity did (i.e., as the inverse square of the distance) includedDaniel Bernoulli^[12] andAlessandro Volta, both of whom measured the force between plates of acapacitor, andFranz Aepinus who supposed the inverse-square law in 1758.^[13]

Based on experiments withelectrically charged spheres,Joseph Priestley of England was among the first to propose that electrical force followed aninverse-square law, similar toNewton's law of universal gravitation. However, he did not generalize or elaborate on this.^[14] In 1767, he conjectured that the force between charges varied as the inverse square of the distance.^[15]^[16]

In 1769, Scottish physicistJohn Robison announced that, according to his measurements, the force of repulsion between two spheres with charges of the same sign varied asx^−2.06.^[17]

In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, byHenry Cavendish of England.^[18] In his notes, Cavendish wrote, "We may therefore conclude that the electric attraction and repulsion must be inversely as some power of the distance between that of the2 +⁠1/50⁠th and that of the2 −⁠1/50⁠th, and there is no reason to think that it differs at all from the inverse duplicate ratio".

Finally, in 1785, the French physicistCharles-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 thetheory of electromagnetism.^[4] He used atorsion balance to study the repulsion and attraction forces ofcharged particles, and determined that the magnitude of the electric force between twopoint charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

The torsion balance consists of a bar suspended from its middle by a thin fiber. The fiber acts as a very weaktorsion spring. In Coulomb's experiment, the torsion balance was aninsulating rod with ametal-coated ball attached to one end, suspended by asilk thread. The ball was charged with a known charge ofstatic electricity, and a second charged ball of the same polarity was brought near it. The two charged balls repelled one another, twisting the fiber through a certain angle, which could be read from a scale on theinstrument. By knowing how much force it took to twist the fiber through a given angle, Coulomb was able to calculate the force between the balls and derive his inverse-square proportionality law.

Mathematical form

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In the image, the vectorF₁ is the force experienced byq₁, and the vectorF₂ is the force experienced byq₂. Whenq₁q₂ > 0 the forces are repulsive (as in the image) and whenq₁q₂ < 0 the forces are attractive (opposite to the image). The magnitude of the forces will always be equal.

Coulomb's law states that the electrostatic force ${\textstyle \mathbf {F} _{1}}$ experienced by a charge, $q_{1}$ at position $\mathbf {r} _{1}$ , in the vicinity of another charge, $q_{2}$ at position $\mathbf {r} _{2}$ , in a vacuum is equal to^[19] $\mathbf {F} _{1}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{12} \over {|\mathbf {r} _{12}|}^{2}}$

where ${\textstyle \mathbf {r_{12}=r_{1}-r_{2}} }$ is thedisplacement vector between the charges, ${\textstyle {\hat {\mathbf {r} }}_{12}}$ aunit vector pointing from ${\textstyle q_{2}}$ to ${\textstyle q_{1}}$ , and $\varepsilon _{0}$ theelectric constant. Here, ${\textstyle \mathbf {\hat {r}} _{12}}$ is used for the vector notation. The electrostatic force ${\textstyle \mathbf {F} _{2}}$ experienced by $q_{2}$ , according toNewton's third law, is ${\textstyle \mathbf {F} _{2}=-\mathbf {F} _{1}}$ .

If both charges have the samesign (like charges) then theproduct $q_{1}q_{2}$ is positive and the direction of the force on $q_{1}$ is given by ${\textstyle {\widehat {\mathbf {r} }}_{12}}$ ; the charges repel each other. If the charges have opposite signs then the product $q_{1}q_{2}$ is negative and the direction of the force on $q_{1}$ is ${\textstyle -{\hat {\mathbf {r} }}_{12}}$ ; the charges attract each other.^[20]

System of discrete charges

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Thelaw of superposition allows Coulomb's law to be extended to include any number of point charges. The force acting on a point charge due to a system of point charges is simply thevector addition of the individual forces acting alone on that point charge due to each one of the charges. The resulting force vector is parallel to theelectric field vector at that point, with that point charge removed.

Force ${\textstyle \mathbf {F} }$ on a small charge $q {\displaystyle q}$ at position $\mathbf {r}$ , due to a system of ${\textstyle n}$ discrete charges in vacuum is^[19]

$\mathbf {F} (\mathbf {r} )={q \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}},$

where $q_{i}$ is the magnitude of theith charge, ${\textstyle \mathbf {r} _{i}}$ is the vector from its position to $\mathbf {r}$ and ${\textstyle {\hat {\mathbf {r} }}_{i}}$ is the unit vector in the direction of $\mathbf {r} _{i}$ .

Continuous charge distribution

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In this case, the principle oflinear superposition is also used. For a continuous charge distribution, anintegral over the region containing the charge is equivalent to an infinite summation, treating eachinfinitesimal element of space as a point charge $d q {\displaystyle dq}$ . The distribution of charge is usually linear, surface or volumetric.

For a linear charge distribution (a good approximation for charge in a wire) where $\lambda (\mathbf {r} ')$ gives the charge per unit length at position $\mathbf {r} '$ , and $d\ell '$ is an infinitesimal element of length,^[21] $dq'=\lambda (\mathbf {r'} )\,d\ell '.$

For a surface charge distribution (a good approximation for charge on a plate in a parallel platecapacitor) where $\sigma (\mathbf {r} ')$ gives the charge per unit area at position $\mathbf {r} '$ , and $d A^{'} {\displaystyle dA'}$ is an infinitesimal element of area, $dq'=\sigma (\mathbf {r'} )\,dA'.$

For a volume charge distribution (such as charge within a bulk metal) where $\rho (\mathbf {r} ')$ gives the charge per unit volume at position $\mathbf {r} '$ , and $d V^{'} {\displaystyle dV'}$ is an infinitesimal element of volume,^[20] $dq'=\rho ({\boldsymbol {r'}})\,dV'.$

The force on a small test charge $q {\displaystyle q}$ at position ${\boldsymbol {r}}$ in vacuum is given by the integral over the distribution of charge $\mathbf {F} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}\int dq'{\frac {\mathbf {r} -\mathbf {r'} }{|\mathbf {r} -\mathbf {r'} |^{3}}}.$

The "continuous charge" version of Coulomb's law is never supposed to be applied to locations for which $|\mathbf {r} -\mathbf {r'} |=0$ because that location would directly overlap with the location of a charged particle (e.g. electron or proton) which is not a valid location to analyze the electric field or potential classically. Charge is always discrete in reality, and the "continuous charge" assumption is just an approximation that is not supposed to allow $|\mathbf {r} -\mathbf {r'} |=0$ to be analyzed.

Coulomb constant

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The constant of proportionality, ${\frac {1}{4\pi \varepsilon _{0}}}$ , in Coulomb's law: $\mathbf {F} _{1}={\frac {q_{1}q_{2}}{4\pi \varepsilon _{0}}}{{\hat {\mathbf {r} }}_{12} \over {|\mathbf {r} _{12}|}^{2}}$ is a consequence of historical choices for units.^[19]^: 4–2

The constant $\varepsilon _{0}$ is thevacuum electric permittivity.^[22] Using theCODATA 2022 recommended value for $\varepsilon _{0}$ ,^[23] the Coulomb constant^[24] is $k_{\text{e}}={\frac {1}{4\pi \varepsilon _{0}}}=8.987\ 551\ 7862(14)\times 10^{9}\ \mathrm {N{\cdot }m^{2}{\cdot }C^{-2}}$

Limitations

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There are three conditions to be fulfilled for the validity of Coulomb's inverse square law:^[25]

The charges must have a spherically symmetric distribution (e.g. be point charges, or a charged metal sphere).
The charges must not overlap (e.g. they must be distinct point charges).
The charges must be stationary with respect to a nonaccelerating frame of reference.

The last of these is known as theelectrostatic approximation. When movement takes place, an extra factor is introduced, which alters the force produced on the two objects. This extra part of the force is called themagnetic force. For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct. A more accurate approximation in this case is, however, theWeber force. When the charges are moving more quickly in relation to each other or accelerations occur,Maxwell's equations andEinstein'stheory of relativity must be taken into consideration.

Electric field

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Main article:Electric field

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

An electric field is avector field that associates to each point in space the Coulomb force experienced by aunit test charge.^[19] The strength and direction of the Coulomb force ${\textstyle \mathbf {F} }$ on a charge ${\textstyle q_{t}}$ depends on the electric field ${\textstyle \mathbf {E} }$ established by other charges that it finds itself in, such that ${\textstyle \mathbf {F} =q_{t}\mathbf {E} }$ . In the simplest case, the field is considered to be generated solely by a single sourcepoint charge. More generally, the field can be generated by a distribution of charges who contribute to the overall by theprinciple of superposition.

If the field is generated by a positive source point charge ${\textstyle q}$ , the direction of the electric field points along lines directed radially outwards from it, i.e. in the direction that a positive point test charge ${\textstyle q_{t}}$ would move if placed in the field. For a negative point source charge, the direction is radially inwards.

The magnitude of the electric fieldE can be derived from Coulomb's law. By choosing one of the point charges to be the source, and the other to be the test charge, it follows from Coulomb's law that the magnitude of theelectric fieldE created by a single sourcepoint chargeQ at a certain distance from itr in vacuum is given by $|\mathbf {E} |=k_{\text{e}}{\frac {|q|}{r^{2}}}$

A system ofn discrete charges $q_{i}$ stationed at ${\textstyle \mathbf {r} _{i}=\mathbf {r} -\mathbf {r} _{i}}$ produces an electric field whose magnitude and direction is, by superposition $\mathbf {E} (\mathbf {r} )={1 \over 4\pi \varepsilon _{0}}\sum _{i=1}^{n}q_{i}{{\hat {\mathbf {r} }}_{i} \over {|\mathbf {r} _{i}|}^{2}}$

Atomic forces

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Relation to Gauss's law

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This articleduplicates the scope of other articles, specificallyGauss's_law#Relation_to_Coulomb's_law. Pleasediscuss this issue and help introduce asummary style to the article.

Deriving Gauss's law from Coulomb's law

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This section is an excerpt fromGauss's law § Deriving Gauss's law from Coulomb's law.[edit]

^{[citation needed]}Strictly speaking, Gauss's law cannot be derived from Coulomb's law alone, since Coulomb's law gives the electric field due to an individual,electrostatic point charge only. However, Gauss's lawcan be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys thesuperposition principle. The superposition principle states that the resulting field is the vector sum of fields generated by each particle (or the integral, if the charges are distributed smoothly in space).

Outline of proof

Coulomb's law states that the electric field due to a stationarypoint charge is: $\mathbf {E} (\mathbf {r} )={\frac {q}{4\pi \varepsilon _{0}}}{\frac {\mathbf {e} _{r}}{r^{2}}}$ where

e_r is the radialunit vector,
r is the radius,|r|,
ε₀ is theelectric constant,
q is the charge of the particle, which is assumed to be located at theorigin.

Using the expression from Coulomb's law, we get the total field atr by using an integral to sum the field atr due to the infinitesimal charge at each other points in space, to give $\mathbf {E} (\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int {\frac {\rho (\mathbf {s} )(\mathbf {r} -\mathbf {s} )}{|\mathbf {r} -\mathbf {s} |^{3}}}\,\mathrm {d} ^{3}\mathbf {s}$ whereρ is the charge density. If we take the divergence of both sides of this equation with respect to r, and use the known theorem^[26]

$\nabla \cdot \left({\frac {\mathbf {r} }{|\mathbf {r} |^{3}}}\right)=4\pi \delta (\mathbf {r} )$ whereδ(r) is theDirac delta function, the result is $\nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {1}{\varepsilon _{0}}}\int \rho (\mathbf {s} )\,\delta (\mathbf {r} -\mathbf {s} )\,\mathrm {d} ^{3}\mathbf {s}$

Using the "sifting property" of the Dirac delta function, we arrive at $\nabla \cdot \mathbf {E} (\mathbf {r} )={\frac {\rho (\mathbf {r} )}{\varepsilon _{0}}},$ which is the differential form of Gauss's law, as desired.

Since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and, in this respect, Gauss's law is more general than Coulomb's law.

Proof (without Dirac Delta)

Let $\Omega \subseteq R^{3}$ be a bounded open set, and $\mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\rho (\mathbf {r} '){\frac {\mathbf {r} -\mathbf {r} '}{\left\|\mathbf {r} -\mathbf {r} '\right\|^{3}}}\mathrm {d} \mathbf {r} '\equiv {\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}$ be the electric field, with $\rho (\mathbf {r} ')$ a continuous function (density of charge).

It is true for all $\mathbf {r} \neq \mathbf {r'}$ that $\nabla _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0$ .

Consider now a compact set $V\subseteq R^{3}$ having apiecewise smooth boundary $\partial V$ such that $\Omega \cap V=\emptyset$ . It follows that $e(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )$ and so, for the divergence theorem:

$\oint _{\partial V}\mathbf {E} _{0}\cdot d\mathbf {S} =\int _{V}\mathbf {\nabla } \cdot \mathbf {E} _{0}\,dV$

But because $e(\mathbf {r,\mathbf {r} '} )\in C^{1}(V\times \Omega )$ ,

$\mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega }\nabla _{\mathbf {r} }\cdot e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}=0$ for the argument above ( $\Omega \cap V=\emptyset \implies \forall \mathbf {r} \in V\ \ \forall \mathbf {r'} \in \Omega \ \ \ \mathbf {r} \neq \mathbf {r'}$ and then $\nabla _{\mathbf {r} }\cdot \mathbf {e} (\mathbf {r,r'} )=0$ )

Therefore the flux through a closed surface generated by some charge density outside (the surface) is null.

Now consider $\mathbf {r} _{0}\in \Omega$ , and $B_{R}(\mathbf {r} _{0})\subseteq \Omega$ as the sphere centered in $\mathbf {r} _{0}$ having $R {\displaystyle R}$ as radius (it exists because $\Omega$ is an open set).

Let $\mathbf {E} _{B_{R}}$ and $\mathbf {E} _{C}$ be the electric field created inside and outside the sphere respectively. Then,

\mathbf {E} _{B_{R}}={\frac {1}{4\pi \varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}

\mathbf {E} _{C}={\frac {1}{4\pi \varepsilon _{0}}}\int _{\Omega \setminus B_{R}(\mathbf {r} _{0})}e(\mathbf {r,\mathbf {r} '} ){\mathrm {d} \mathbf {r} '}

and

\mathbf {E} _{B_{R}}+\mathbf {E} _{C}=\mathbf {E} _{0}

$\Phi (R)=\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{0}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S} +\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{C}\cdot d\mathbf {S} =\oint _{\partial B_{R}(\mathbf {r} _{0})}\mathbf {E} _{B_{R}}\cdot d\mathbf {S}$

The last equality follows by observing that $(\Omega \setminus B_{R}(\mathbf {r} _{0}))\cap B_{R}(\mathbf {r} _{0})=\emptyset$ , and the argument above.

The RHS is the electric flux generated by a charged sphere, and so:

$\Phi (R)={\frac {Q(R)}{\varepsilon _{0}}}={\frac {1}{\varepsilon _{0}}}\int _{B_{R}(\mathbf {r} _{0})}\rho (\mathbf {r} '){\mathrm {d} \mathbf {r} '}={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} '_{c})|B_{R}(\mathbf {r} _{0})|$ with $r'_{c}\in \ B_{R}(\mathbf {r} _{0})$

Where the last equality follows by the mean value theorem for integrals. Using thesqueeze theorem and the continuity of $\rho$ , one arrives at:

$\mathbf {\nabla } \cdot \mathbf {E} _{0}(\mathbf {r} _{0})=\lim _{R\to 0}{\frac {1}{|B_{R}(\mathbf {r} _{0})|}}\Phi (R)={\frac {1}{\varepsilon _{0}}}\rho (\mathbf {r} _{0})$

Deriving Coulomb's law from Gauss's law

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Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding thecurl ofE (seeHelmholtz decomposition andFaraday's law). However, Coulomb's lawcan be proven from Gauss's law if it is assumed, in addition, that the electric field from apoint charge is spherically symmetric (this assumption, like Coulomb's law itself, is exactly true if the charge is stationary, and approximately true if the charge is in motion).

Outline of proof

TakingS in the integral form of Gauss's law to be a spherical surface of radiusr, centered at the point chargeQ, we have $\oint _{S}\mathbf {E} \cdot d\mathbf {A} ={\frac {Q}{\varepsilon _{0}}}$

By the assumption of spherical symmetry, the integrand is a constant which can be taken out of the integral. The result is $4\pi r^{2}{\hat {\mathbf {r} }}\cdot \mathbf {E} (\mathbf {r} )={\frac {Q}{\varepsilon _{0}}}$ wherer̂ is aunit vector pointing radially away from the charge. Again by spherical symmetry,E points in the radial direction, and so we get $\mathbf {E} (\mathbf {r} )={\frac {Q}{4\pi \varepsilon _{0}}}{\frac {\hat {\mathbf {r} }}{r^{2}}}$ which is essentially equivalent to Coulomb's law. Thus theinverse-square law dependence of the electric field in Coulomb's law follows from Gauss's law.

In relativity

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Coulomb's law can be used to gain insight into the form of the magnetic field generated by moving charges since by special relativity, in certain cases themagnetic field can be shown to be a transformation of forces caused by theelectric field. When no acceleration is involved in a particle's history, Coulomb's law can be assumed on any test particle in its own inertial frame, supported by symmetry arguments in solvingMaxwell's equation, shown above. Coulomb's law can be expanded to moving test particles to be of the same form. This assumption is supported byLorentz force law which, unlike Coulomb's law is not limited to stationary test charges. Considering the charge to be invariant of observer, the electric and magnetic fields of a uniformly moving point charge can hence be derived by theLorentz transformation of thefour force on the test charge in the charge's frame of reference given by Coulomb's law and attributing magnetic and electric fields by their definitions given by the form ofLorentz force.^[27] The fields hence found for uniformly moving point charges are given by:^[28] $\mathbf {E} ={\frac {q}{4\pi \epsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}\mathbf {r}$ $\mathbf {B} ={\frac {q}{4\pi \epsilon _{0}r^{3}}}{\frac {1-\beta ^{2}}{(1-\beta ^{2}\sin ^{2}\theta )^{3/2}}}{\frac {\mathbf {v} \times \mathbf {r} }{c^{2}}}={\frac {\mathbf {v} \times \mathbf {E} }{c^{2}}}$ where $q {\displaystyle q}$ is the charge of the point source, $\mathbf {r}$ is the position vector from the point source to the point in space, $\mathbf {v}$ is the velocity vector of the charged particle, $\beta$ is the ratio of speed of the charged particle divided by the speed of light and $\theta$ is the angle between $\mathbf {r}$ and $\mathbf {v}$ .

This form of solutions need not obeyNewton's third law as is the case in the framework ofspecial relativity (yet without violating relativistic-energy momentum conservation).^[29] Note that the expression for electric field reduces to Coulomb's law for non-relativistic speeds of the point charge and that the magnetic field in non-relativistic limit (approximating $\beta \ll 1$ ) can be applied to electric currents to get theBiot–Savart law. These solutions, when expressed in retarded time also correspond to the general solution ofMaxwell's equations given by solutions ofLiénard–Wiechert potential, due to the validity of Coulomb's law within its specific range of application. Also note that the spherical symmetry for gauss law on stationary charges is not valid for moving charges owing to the breaking of symmetry by the specification of direction of velocity in the problem. Agreement withMaxwell's equations can also be manually verified for the above two equations.^[30]

Coulomb potential

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Quantum field theory

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The most basic Feynman diagram for QED interaction between two fermions

TheCoulomb potential admits continuum states (withE > 0), describing electron-protonscattering, as well as discrete bound states, representing the hydrogen atom.^[31] It can also be derived within thenon-relativistic limit between two charged particles, as follows:

UnderBorn approximation, in non-relativistic quantum mechanics, the scattering amplitude ${\textstyle {\mathcal {A}}(|\mathbf {p} \rangle \to |\mathbf {p} '\rangle )}$ is: ${\mathcal {A}}(|\mathbf {p} \rangle \to |\mathbf {p} '\rangle )-1=2\pi \delta (E_{p}-E_{p'})(-i)\int d^{3}\mathbf {r} \,V(\mathbf {r} )e^{-i(\mathbf {p} -\mathbf {p} ')\mathbf {r} }$ This is to be compared to the: $\int {\frac {d^{3}k}{(2\pi )^{3}}}e^{ikr_{0}}\langle p',k|S|p,k\rangle$ where we look at the (connected) S-matrix entry for two electrons scattering off each other, treating one with "fixed" momentum as the source of the potential, and the other scattering off that potential.

Using the Feynman rules to compute the S-matrix element, we obtain in the non-relativistic limit with $m_{0}\gg |\mathbf {p} |$ $\langle p',k|S|p,k\rangle |_{conn}=-i{\frac {e^{2}}{|\mathbf {p} -\mathbf {p} '|^{2}-i\varepsilon }}(2m)^{2}\delta (E_{p,k}-E_{p',k})(2\pi )^{4}\delta (\mathbf {p} -\mathbf {p} ')$

Comparing with the QM scattering, we have to discard the $(2m)^{2}$ as they arise due to differing normalizations of momentum eigenstate in QFT compared to QM and obtain: $\int V(\mathbf {r} )e^{-i(\mathbf {p} -\mathbf {p} ')\mathbf {r} }d^{3}\mathbf {r} ={\frac {e^{2}}{|\mathbf {p} -\mathbf {p} '|^{2}-i\varepsilon }}$ where Fourier transforming both sides, solving the integral and taking $\varepsilon \to 0$ at the end will yield $V(r)={\frac {e^{2}}{4\pi r}}$ as the Coulomb potential.^[32]

However, the equivalent results of the classical Born derivations for the Coulomb problem are thought to be strictly accidental.^[33]^[34]

The Coulomb potential, and its derivation, can be seen as a special case of theYukawa potential, which is the case where the exchanged boson – the photon – has no rest mass.^[31]

Verification

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It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass $m {\displaystyle m}$ and same-sign charge $q {\displaystyle q}$ , hanging from two ropes of negligible mass of length $l {\displaystyle l}$ . The forces acting on each sphere are three: the weight $m g {\displaystyle mg}$ , the rope tension $\mathbf {T}$ and the electric force $\mathbf {F}$ . In the equilibrium state:

\mathbf {T} \sin \theta _{1}=\mathbf {F} _{1}

and

\mathbf {T} \cos \theta _{1}=mg

Dividing (1) by (2):

{\frac {\sin \theta _{1}}{\cos \theta _{1}}}={\frac {F_{1}}{mg}}\Rightarrow F_{1}=mg\tan \theta _{1}

Let $\mathbf {L} _{1}$ be the distance between the charged spheres; the repulsion force between them $\mathbf {F} _{1}$ , assuming Coulomb's law is correct, is equal to

F_{1}={\frac {q^{2}}{4\pi \varepsilon _{0}L_{1}^{2}}}

Coulomb's law

so:

{\frac {q^{2}}{4\pi \varepsilon _{0}L_{1}^{2}}}=mg\tan \theta _{1}

If we now discharge one of the spheres, and we put it in contact with the charged sphere, each one of them acquires a charge ${\textstyle {\frac {q}{2}}}$ . In the equilibrium state, the distance between the charges will be ${\textstyle \mathbf {L} _{2}<\mathbf {L} _{1}}$ and the repulsion force between them will be:

F_{2}={\frac {{({\frac {q}{2}})}^{2}}{4\pi \varepsilon _{0}L_{2}^{2}}}={\frac {\frac {q^{2}}{4}}{4\pi \varepsilon _{0}L_{2}^{2}}}

We know that $\mathbf {F} _{2}=mg\tan \theta _{2}$ and: ${\frac {\frac {q^{2}}{4}}{4\pi \varepsilon _{0}L_{2}^{2}}}=mg\tan \theta _{2}$ Dividing (4) by (5), we get:

{\frac {\left({\cfrac {q^{2}}{4\pi \varepsilon _{0}L_{1}^{2}}}\right)}{\left({\cfrac {\frac {q^{2}}{4}}{4\pi \varepsilon _{0}L_{2}^{2}}}\right)}}={\frac {mg\tan \theta _{1}}{mg\tan \theta _{2}}}\Rightarrow 4{\left({\frac {L_{2}}{L_{1}}}\right)}^{2}={\frac {\tan \theta _{1}}{\tan \theta _{2}}}

Measuring the angles $\theta _{1}$ and $\theta _{2}$ and the distance between the charges $\mathbf {L} _{1}$ and $\mathbf {L} _{2}$ is sufficient 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 great, the angles will be small enough to make the following approximation:

\tan \theta \approx \sin \theta ={\frac {\frac {L}{2}}{\ell }}={\frac {L}{2\ell }}\Rightarrow {\frac {\tan \theta _{1}}{\tan \theta _{2}}}\approx {\frac {\frac {L_{1}}{2\ell }}{\frac {L_{2}}{2\ell }}}

Using this approximation, the relationship (6) becomes the much simpler expression:

{\frac {\frac {L_{1}}{2\ell }}{\frac {L_{2}}{2\ell }}}\approx 4{\left({\frac {L_{2}}{L_{1}}}\right)}^{2}\Rightarrow {\frac {L_{1}}{L_{2}}}\approx 4{\left({\frac {L_{2}}{L_{1}}}\right)}^{2}\Rightarrow {\frac {L_{1}}{L_{2}}}\approx {\sqrt[{3}]{4}}

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

References

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^^a ^bGriffiths, David J. (16 August 2018).Introduction to quantum mechanics (Third ed.). Cambridge, United Kingdom.ISBN 978-1-107-18963-8.{{cite book}}: CS1 maint: location missing publisher (link)
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Spavieri, G., Gillies, G. T., & Rodriguez, M. (2004). Physical implications of Coulomb’s Law. Metrologia, 41(5), S159–S170. doi:10.1088/0026-1394/41/5/s06

External links

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Wikimedia Commons has media related toCoulomb's law.

Coulomb's Law onProject PHYSNET
Electricity and the Atom Archived 2009-02-21 at theWayback Machine—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.