# Coulomb's law

Last updated

Coulomb's law, or Coulomb's inverse-square law, is an experimental law [1] of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is conventionally called electrostatic force [2] or Coulomb force. [3] The quantity of electrostatic force between stationary charges is always described by Coulomb's law. [4] The law was first published in 1785 by French physicist Charles-Augustin de Coulomb, and was essential to the development of the theory of electromagnetism, maybe even its starting point, [5] because it was now possible to discuss quantity of electric charge in a meaningful way. [6]

Physics is the natural science that studies matter, its motion and behavior through space and time, and that studies the related entities of energy and force. Physics is one of the most fundamental scientific disciplines, and its main goal is to understand how the universe behaves.

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: positive and negative. Like charges repel and unlike attract. An object with an absence of net charge is referred to as neutral. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.

Charles-Augustin de Coulomb was a French military engineer and physicist. He is best known as the eponymous discoverer of what is now called Coulomb's law, the description of the electrostatic force of attraction and repulsion, though he also did important work on friction.

## Contents

In its scalar form, the law is:

${\displaystyle F=k_{e}{\frac {q_{1}q_{2}}{r^{2}}},}$

where ke is Coulomb's constant (ke9×109 N⋅m2⋅C−2), [7] q1 and q2 are the signed magnitudes of the charges, and the scalar r is the distance between the charges. The force of the interaction between the charges is attractive if the charges have opposite signs (i.e., F is negative) and repulsive if like-signed (i.e., F is positive).

Being an inverse-square law, the law is analogous to Isaac Newton's inverse-square law of universal gravitation, but gravitational forces are always attractive, while electrostatic forces can be attractive or repulsive. [8] Coulomb's law can be used to derive Gauss's law, and vice versa. In the case of a single stationary point charge, the two laws are equivalent, expressing the same physical law in different ways. [9] The law has been tested extensively, and observations have upheld the law on a scale from 10−16 m to 108 m. [10]

The inverse-square law, in physics, is any physical law stating that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.

Sir Isaac Newton was an English mathematician, physicist, astronomer, theologian, and author who is widely recognised as one of the most influential scientists of all time, and a key figure in the scientific revolution. His book Philosophiæ Naturalis Principia Mathematica, first published in 1687, laid the foundations of classical mechanics. Newton also made seminal contributions to optics, and shares credit with Gottfried Wilhelm Leibniz for developing the infinitesimal calculus.

Newton's law of universal gravitation states that every particle attracts every other particle in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This is a general physical law derived from empirical observations by what Isaac Newton called inductive reasoning. It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica, first published on 5 July 1687. When Newton presented Book 1 of the unpublished text in April 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him.

## History

Ancient cultures around the Mediterranean knew that certain objects, such as rods of amber, could be rubbed with cat's fur to attract light objects like feathers and papers. Thales of Miletus made a series of observations on static electricity around 600 BC, from which he believed that friction rendered amber magnetic, in contrast to minerals such as magnetite, which needed no rubbing. [11] [12] Thales was incorrect in believing the attraction was due to a magnetic effect, but later science would prove a link between magnetism and electricity. Electricity would remain little more than an intellectual curiosity for millennia until 1600, when the English scientist William Gilbert made a careful study of electricity and magnetism, distinguishing the lodestone effect from static electricity produced by rubbing amber. [11] He coined the New Latin word electricus ("of amber" or "like amber", from ἤλεκτρον [elektron], the Greek word for "amber") to refer to the property of attracting small objects after being rubbed. [13] This association gave rise to the English words "electric" and "electricity", which made their first appearance in print in Thomas Browne's Pseudodoxia Epidemica of 1646. [14]

The Mediterranean Sea is a sea connected to the Atlantic Ocean, surrounded by the Mediterranean Basin and almost completely enclosed by land: on the north by Southern Europe and Anatolia, on the south by North Africa and on the east by the Levant. Although the sea is sometimes considered a part of the Atlantic Ocean, it is usually referred to as a separate body of water. Geological evidence indicates that around 5.9 million years ago, the Mediterranean was cut off from the Atlantic and was partly or completely desiccated over a period of some 600,000 years before being refilled by the Zanclean flood about 5.3 million years ago.

Amber is fossilized tree resin, which has been appreciated for its color and natural beauty since Neolithic times. Much valued from antiquity to the present as a gemstone, amber is made into a variety of decorative objects. Amber is used in jewelry. It has also been used as a healing agent in folk medicine.

Thales of Miletus was a Greek mathematician, astronomer and pre-Socratic philosopher from Miletus in ancient Greek Ionia. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him as the first philosopher in the Greek tradition, and he is otherwise historically recognized as the first individual in Western civilization known to have entertained and engaged in scientific philosophy.

Early investigators of the 18th century who suspected that the electrical force diminished with distance as the force of gravity did (i.e., as the inverse square of the distance) included Daniel Bernoulli [15] and Alessandro Volta, both of whom measured the force between plates of a capacitor, and Franz Aepinus who supposed the inverse-square law in 1758. [16]

In physics, a force is any interaction that, when unopposed, will change the motion of an object. A force can cause an object with mass to change its velocity, i.e., to accelerate. Force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity. It is measured in the SI unit of newtons and represented by the symbol F.

Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward one another. On Earth, gravity gives weight to physical objects, and the Moon's gravity causes the ocean tides. The gravitational attraction of the original gaseous matter present in the Universe caused it to begin coalescing, forming stars—and for the stars to group together into galaxies—so gravity is responsible for many of the large-scale structures in the Universe. Gravity has an infinite range, although its effects become increasingly weaker on farther objects.

Daniel Bernoulli FRS was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics. His name is commemorated in the Bernoulli's principle, a particular example of the conservation of energy, which describes the mathematics of the mechanism underlying the operation of two important technologies of the 20th century: the carburetor and the airplane wing.

Based on experiments with electrically 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. [17] In 1767, he conjectured that the force between charges varied as the inverse square of the distance. [18] [19]

Joseph Priestley was an 18th-century English Separatist theologian, natural philosopher, chemist, innovative grammarian, multi-subject educator, and liberal political theorist who published over 150 works. He has historically been credited with the discovery of oxygen, having isolated it in its gaseous state, although Carl Wilhelm Scheele and Antoine Lavoisier also have strong claims to the discovery.

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. [20]

In the early 1770s, the dependence of the force between charged bodies upon both distance and charge had already been discovered, but not published, by Henry Cavendish of England. [21]

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. [22] 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 torsion balance consists of a bar suspended from its middle by a thin fiber. The fiber acts as a very weak torsion spring. In Coulomb's experiment, the torsion balance was an insulating rod with a metal-coated ball attached to one end, suspended by a silk thread. The ball was charged with a known charge of static 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 the instrument. 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.

## The law

Coulomb's law states that:

The magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. [22]

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

Coulomb's law can also be stated as a simple mathematical expression. The scalar and vector forms of the mathematical equation are

${\displaystyle |\mathbf {F} |=k_{e}{|q_{1}q_{2}| \over r^{2}}\qquad }$ and ${\displaystyle \qquad \mathbf {F} _{1}=k_{e}{\frac {q_{1}q_{2}}{{|\mathbf {r} _{21}|}^{2}}}\mathbf {\widehat {r}} _{21},\qquad }$ respectively,

where ke is Coulomb's constant (ke = 8.9875517873681764×109 N⋅m2⋅C−2), q1 and q2 are the signed magnitudes of the charges, the scalar r is the distance between the charges, the vector r21 = r1r2 is the vectorial distance between the charges, and 21 = r21/|r21| (a unit vector pointing from q2 to q1). The vector form of the equation calculates the force F1 applied on q1 by q2. If r12 is used instead, then the effect on q2 can be found. It can be also calculated using Newton's third law: F2 = −F1.

### Units

When the electromagnetic theory is expressed in the International System of Units, force is measured in newtons, charge in coulombs, and distance in meters. Coulomb's constant is given by ke = 1/ε0. The constant ε0 is the vacuum electric permittivity (also known as "electric constant") [23] in C2⋅m−2⋅N−1. It should not be confused with εr, which is the dimensionless relative permittivity of the material in which the charges are immersed, or with their product εa = ε0εr, which is called "absolute permittivity of the material" and is still used in electrical engineering.

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

Coulomb's law and Coulomb's constant can also be interpreted in various terms:

Gaussian units and Lorentz–Heaviside units are both CGS unit systems. Gaussian units are more amenable for microscopic problems such as the electrodynamics of individual electrically charged particles. [24] SI units are more convenient for practical, large-scale phenomena, such as engineering applications. [24]

### Electric field

An electric field is a vector field that associates to each point in space the Coulomb force experienced by a test charge. In the simplest case, the field is considered to be generated solely by a single source point charge. The strength and direction of the Coulomb force F on a test charge qt depends on the electric field E that it finds itself in, such that F = qtE. If the field is generated by a positive source point charge 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 qt would move if placed in the field. For a negative point source charge, the direction is radially inwards.

The magnitude of the electric field E 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 the electric field E created by a single source point charge q at a certain distance from it r in vacuum is given by:

${\displaystyle |{\boldsymbol {E}}|={1 \over 4\pi \varepsilon _{0}}{|q| \over r^{2}}}$

### Coulomb's constant

Coulomb's constant is a proportionality factor that appears in Coulomb's law as well as in other electric-related formulas. The value of this constant is dependent upon the medium that the charged objects are immersed in. Denoted ke, it is also called the electric force constant or electrostatic constant, [8] hence the subscript e.

The exact value of Coulomb's constant in the case of air or vacuum is:

{\displaystyle {\begin{aligned}k_{e}&={\frac {1}{4\pi \varepsilon _{0}}}={\frac {c_{0}^{2}\mu _{0}}{4\pi }}=c_{0}^{2}\times 10^{-7}\ \mathrm {H\cdot m} ^{-1}\\&=8.987\,551\,787\,368\,176\,4\times 10^{9}\ \mathrm {N\cdot m^{2}\cdot C} ^{-2}\end{aligned}}}

### Limitations

There are three conditions to be fulfilled for the validity of Coulomb's law:

1. The charges must have a spherically symmetric distribution (e.g. be point charges, or a charged metal sphere).
2. The charges must not overlap (e.g. they must be distinct point charges).
3. The charges must be stationary with respect to each other.

The last of these is known as the electrostatic approximation. When movement takes place, Einstein's theory of relativity must be taken into consideration, and a result, an extra factor is introduced, which alters the force produced on the two objects. This extra part of the force is called the magnetic force, and is described by magnetic fields. For slow movement, the magnetic force is minimal and Coulomb's law can still be considered approximately correct, but when the charges are moving more quickly in relation to each other, the full electrodynamic rules (incorporating the magnetic force) must be considered.

### Quantum field theory origin

In simple terms, the Coulomb potential derives from the QED Lagrangian as follows. The Lagrangian of quantum electrodynamics is normally written in natural units, but in SI units, it is:

${\displaystyle {\mathcal {L}}_{\mathrm {QED} }={\bar {\psi }}(i\hbar c\gamma ^{\mu }D_{\mu }-mc^{2})\psi -{\frac {1}{4\hbar c}}F_{\mu \nu }F^{\mu \nu }}$

where the covariant derivative (in SI units) is:

${\displaystyle {D}_{\mu }=\partial _{\mu }+{\frac {ig}{\hbar c}}A_{\mu }}$

where ${\displaystyle g}$ is the gauge coupling parameter. By putting the covariant derivative into the lagrangian explicitly, the interaction term (the term involving both ${\displaystyle A}$ and ${\displaystyle \psi }$) is seen to be:

${\displaystyle {\mathcal {L}}_{\mathrm {int} }=ig{\bar {\psi }}\gamma ^{\mu }A_{\mu }\psi }$

The most basic Feynman diagram for a QED interaction between two fermions is the exchange of a single photon, with no loops. Following the Feynman rules, this therefore contributes two QED vertex factors (${\displaystyle igQ\gamma _{\mu }}$) to the potential, where Q is the QED-charge operator (Q gives the charge in terms of the electron charge, and hence is exactly −1 for electrons, etc.). For the photon in the diagram, the Feynman rules demand the contribution of one bosonic massless propagator ${\displaystyle \left({\frac {\hbar c}{k^{2}}}\right)}$. Ignoring the momentum on the external legs (the fermions), the potential is therefore:

${\displaystyle V(\mathbf {r} )={\frac {1}{(2\pi )^{3}}}\int e^{\frac {i\mathbf {k\cdot r} }{\hbar }}(iQ_{1}g^{2}\gamma _{\mu })(iQ_{2}g^{2}\gamma _{\nu })\left({\frac {\hbar c}{k^{2}}}\right)\;d^{3}k}$

which can be more usefully written as

${\displaystyle V(\mathbf {r} )={\frac {-g^{2}\hbar cQ_{1}Q_{2}}{4\pi }}{\frac {1}{(2\pi )^{3}}}\int e^{\frac {i\mathbf {k\cdot r} }{\hbar }}{\frac {4\pi \eta _{\mu \nu }}{k^{2}}}\;d^{3}k}$

where ${\displaystyle Q_{i}}$ is the QED-charge on the ith particle. Recognising the integral as just being a Fourier transform enables the equation to be simplified:

${\displaystyle V(\mathbf {r} )={\frac {-g^{2}\hbar cQ_{1}Q_{2}}{4\pi }}{\frac {1}{r}}.}$

For various reasons, it is more convenient to define the fine-structure constant ${\displaystyle \alpha ={\frac {g^{2}}{4\pi }}}$, and then define ${\displaystyle e={\sqrt {4\pi \alpha \varepsilon _{0}\hbar c}}}$. Rearranging these definitions gives:

${\displaystyle g^{2}\hbar c={\frac {e^{2}}{\varepsilon _{0}}}}$

In natural units (${\displaystyle \hbar =1}$, ${\displaystyle c=1}$, and ${\displaystyle \varepsilon _{0}=1}$), ${\displaystyle g=e}$. Continuing in SI units, the potential is therefore

${\displaystyle V(\mathbf {r} )={\frac {-e^{2}}{4\pi \varepsilon _{0}}}{\frac {Q_{1}Q_{2}}{r}}}$

Defining ${\displaystyle q_{i}=eQ_{i}}$, as the macroscopic 'electric charge', makes e the macroscopic 'electric charge' for an electron, and enables the formula to be put into the familiar form of the Coulomb potential:

${\displaystyle V(\mathbf {r} )={\frac {-1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r}}}$

The force (${\displaystyle {\frac {dV(\mathbf {r} )}{dr}}}$) is therefore :

${\displaystyle F(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}$

The derivation makes clear that the force law is only an approximation — it ignores the momentum of the input and output fermion lines, and ignores all quantum corrections (i.e. the myriad possible diagrams with internal loops).

The Coulomb potential, and its derivation, can be seen as a special case of the Yukawa potential (specifically, the case where the exchanged boson – the photon – has no rest mass).

## Scalar form

When it is of interest to know the magnitude of the electrostatic force (and not its direction), it may be easiest to consider a scalar version of the law. The scalar form of Coulomb's Law relates the magnitude and sign of the electrostatic force F acting simultaneously on two point charges q1 and q2 as follows:

${\displaystyle |{\boldsymbol {F}}|=k_{e}{|q_{1}q_{2}| \over r^{2}}}$

where r is the separation distance and ke is Coulomb's constant. If the product q1q2 is positive, the force between the two charges is repulsive; if the product is negative, the force between them is attractive. [25]

## Vector form

Coulomb's law states that the electrostatic force F1 experienced by a charge, q1 at position r1, in the vicinity of another charge, q2 at position r2, in a vacuum is equal to:

${\displaystyle {\boldsymbol {F_{1}}}={q_{1}q_{2} \over 4\pi \varepsilon _{0}}{({\boldsymbol {r_{1}-r_{2}}}) \over |{\boldsymbol {r_{1}-r_{2}}}|^{3}}={q_{1}q_{2} \over 4\pi \varepsilon _{0}}{{\boldsymbol {{\widehat {r}}_{21}}} \over |{\boldsymbol {r_{21}}}|^{2}},}$

where r21 = r1r2, the unit vector 21 = r21/|r21|, and ε0 is the electric constant.

The vector form of Coulomb's law is simply the scalar definition of the law with the direction given by the unit vector, 21, parallel with the line from charge q2to charge q1. [26] If both charges have the same sign (like charges) then the product q1q2 is positive and the direction of the force on q1 is given by 21; the charges repel each other. If the charges have opposite signs then the product q1q2 is negative and the direction of the force on q1 is given by 21 = 12; the charges attract each other.

The electrostatic force F2 experienced by q2, according to Newton's third law, is F2 = −F1.

### System of discrete charges

The law 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 the vector 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 the electric field vector at that point, with that point charge removed.

The force F on a small charge q at position r, due to a system of N discrete charges in vacuum is:

${\displaystyle {\boldsymbol {F(r)}}={q \over 4\pi \varepsilon _{0}}\sum _{i=1}^{N}q_{i}{{\boldsymbol {r-r_{i}}} \over |{\boldsymbol {r-r_{i}}}|^{3}}={q \over 4\pi \varepsilon _{0}}\sum _{i=1}^{N}q_{i}{{\boldsymbol {\widehat {R_{i}}}} \over |{\boldsymbol {R_{i}}}|^{2}},}$

where qi and ri are the magnitude and position respectively of the ith charge, i is a unit vector in the direction of Ri = rri (a vector pointing from charges qi to q). [26]

### Continuous charge distribution

In this case, the principle of linear superposition is also used. For a continuous 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. The distribution of charge is usually linear, surface or volumetric.

For a linear charge distribution (a good approximation for charge in a wire) where λ(r′) gives the charge per unit length at position r′, and dℓ′ is an infinitesimal element of length,

${\displaystyle dq=\lambda ({\boldsymbol {r'}})\,d\ell '.}$ [27]

For a surface charge distribution (a good approximation for charge on a plate in a parallel plate capacitor) where σ(r′) gives the charge per unit area at position r′, and dA′ is an infinitesimal element of area,

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

For a volume charge distribution (such as charge within a bulk metal) where ρ(r′) gives the charge per unit volume at position r′, and dV′ is an infinitesimal element of volume,

${\displaystyle dq=\rho ({\boldsymbol {r'}})\,dV'.}$ [26]

The force on a small test charge q′ at position r in vacuum is given by the integral over the distribution of charge:

${\displaystyle {\boldsymbol {F}}={q' \over 4\pi \varepsilon _{0}}\int dq{{\boldsymbol {r}}-{\boldsymbol {r'}} \over |{\boldsymbol {r}}-{\boldsymbol {r'}}|^{3}}.}$

## Simple experiment to verify Coulomb's law

It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass m and same-sign charge q, hanging from two ropes of negligible mass of 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:

${\displaystyle T\ \sin \theta _{1}=F_{1}\,\!}$

(1)

and:

${\displaystyle T\ \cos \theta _{1}=mg\,\!}$

(2)

Dividing ( 1 ) by ( 2 ):

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

(3)

Let L1 be the distance between the charged spheres; the repulsion force between them F1, assuming Coulomb's law is correct, is equal to

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

(Coulomb's law)

so:

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

(4)

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 q/2. In the equilibrium state, the distance between the charges will be L2 < L1 and the repulsion force between them will be:

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

(5)

We know that F2 = mg tan θ2. And:

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

Dividing ( 4 ) by ( 5 ), we get:

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

(6)

Measuring the angles θ1 and θ2 and the distance between the charges L1 and L2 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:

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

(7)

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

${\displaystyle {\frac {\frac {L_{1}}{2\ell }}{\frac {L_{2}}{2\ell }}}\approx 4{\left({\frac {L_{2}}{L_{1}}}\right)}^{2}\Rightarrow \,\!}$${\displaystyle {\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}}\,\!}$

(8)

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

## Atomic forces

Coulomb's law holds even within atoms, correctly describing the force between the positively charged atomic 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 force of attraction, and binding energy, approach zero and ionic bonding is less favorable. As the magnitude of opposing charges increases, energy increases and ionic bonding is more favorable.

## Relation to Gauss's law

### Deriving Gauss's law from Coulomb's law

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 point charge only. However, Gauss's law can be proven from Coulomb's law if it is assumed, in addition, that the electric field obeys the superposition principle. The superposition principle says 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).

Note that 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.

### Deriving Coulomb's law from Gauss's law

Strictly speaking, Coulomb's law cannot be derived from Gauss's law alone, since Gauss's law does not give any information regarding the curl of E (see Helmholtz decomposition and Faraday's law). However, Coulomb's law can be proven from Gauss's law if it is assumed, in addition, that the electric field from a point 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).

## Notes

1. Huray 2010 , p. 57
2. Jackson 1999 , p. 24
3. Huray 2010 , p. 2
4. Roller, Duane; Roller, D.H.D. (1954). The development of the concept of electric charge: Electricity from the Greeks to Coulomb. Cambridge, MA: Harvard University Press. p. 79.
5. Huray 2010 , p. 7
6. Stewart, Joseph (2001). Intermediate Electromagnetic Theory. World Scientific. p. 50. ISBN   978-981-02-4471-2
7. Simpson, Brian (2003). Electrical Stimulation and the Relief of Pain. Elsevier Health Sciences. pp. 6–7. ISBN   978-0-444-51258-1
8. Baigrie, Brian (2007). Electricity and Magnetism: A Historical Perspective. Greenwood Press. pp. 7–8. ISBN   978-0-313-33358-3
9. Chalmers, Gordon (1937). "The Lodestone and the Understanding of Matter in Seventeenth Century England". Philosophy of Science. 4 (1): 75–95. doi:10.1086/286445
10. Socin, Abel (1760). Acta Helvetica Physico-Mathematico-Anatomico-Botanico-Medica (in Latin). 4. Basileae. pp. 224–25.
11. Heilbron, J.L. (1979). Electricity in the 17th and 18th Centuries: A Study of Early Modern Physics. Los Angeles, California: University of California Press. pp.  460–462 and 464 (including footnote 44). ISBN   978-0486406886.
12. Schofield, Robert E. (1997). The Enlightenment of Joseph Priestley: A Study of his Life and Work from 1733 to 1773. University Park: Pennsylvania State University Press. pp. 144–56. ISBN   978-0-271-01662-7.
13. Priestley, Joseph (1767). The History and Present State of Electricity, with Original Experiments. London, England. p. 732.
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?
14. Elliott, Robert S. (1999). Electromagnetics: History, Theory, and Applications. ISBN   978-0-7803-5384-8.
15. Robison, John (1822). Murray, John (ed.). A System of Mechanical Philosophy. 4. London, England.
On page 68, the author states that in 1769 he announced his findings regarding the force between spheres of like charge. On page 73, the author states the force between spheres of like charge varies as x−2.06:
The result of the whole was, that the mutual repulsion of two spheres, electrified positively or negatively, was very nearly in the inverse proportion of the squares of the distances of their centres, or rather in a proportion somewhat greater, approaching to x−2.06.
When making experiments with charged spheres of opposite charge the results were similar, as stated on page 73:
When the experiments were repeated with balls having opposite electricities, and which therefore attracted each other, the results were not altogether so regular and a few irregularities amounted to 16 of the whole; but these anomalies were as often on one side of the medium as on the other. This series of experiments gave a result which deviated as little as the former (or rather less) from the inverse duplicate ratio of the distances; but the deviation was in defect as the other was in excess.
Nonetheless, on page 74 the author infers that the actual action is related exactly to the inverse duplicate of the distance:
We therefore think that it may be concluded, that the action between two spheres is exactly in the inverse duplicate ratio of the distance of their centres, and that this difference between the observed attractions and repulsions is owing to some unperceived cause in the form of the experiment.
On page 75, the authour compares the electric and gravitational forces:
Therefore we may conclude, that the law of electric attraction and repulsion is similar to that of gravitation, and that each of those forces diminishes in the same proportion that the square of the distance between the particles increases.
16. Maxwell, James Clerk, ed. (1967) [1879]. "Experiments on Electricity: Experimental determination of the law of electric force.". The Electrical Researches of the Honourable Henry Cavendish... (1st ed.). Cambridge, England: Cambridge University Press. pp. 104–113.
On pages 111 and 112 the author states:
We may therefore conclude that the electric attraction and repulsion must be inversely as some power of the distance between that of the 2 + 150 th and that of the 2 − 150 th, and there is no reason to think that it differs at all from the inverse duplicate ratio.
17. Coulomb (1785a) "Premier mémoire sur l’électricité et le magnétisme," Histoire de l’Académie Royale des Sciences, pp. 569–577 — Coulomb studied the repulsive force between bodies having electrical charges of the same sign:
Il résulte donc de ces trois essais, que l'action répulsive que les deux balles électrifées de la même nature d'électricité exercent l'une sur l'autre, suit la raison inverse du carré des distances. Translation: 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.
Coulomb also showed that oppositely charged bodies obey an inverse-square law of attraction.
18. International Bureau of Weights and Measures (2018-02-05), SI Brochure: The International System of Units (SI) (PDF) (Draft) (9th ed.), p. 15
19. Jackson 1999 , p. 784
20. Coulomb's law, Hyperphysics
21. Coulomb's law, University of Texas
22. Charged rods, PhysicsLab.org
23. See, for example, Griffiths, David J. (2013). Introduction to Electrodynamics (4th ed.). Prentice Hall. p. 50.

## Related Research Articles

The centimetre–gram–second system of units is a variant of the metric system based on the centimetre as the unit of length, the gram as the unit of mass, and the second as the unit of time. All CGS mechanical units are unambiguously derived from these three base units, but there are several different ways of extending the CGS system to cover electromagnetism.

In quantum mechanics, a Hamiltonian is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , but also or to highlight its function as an operator. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strength. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.

In physics, specifically electromagnetism, the Biot–Savart law is an equation describing the magnetic field generated by a constant electric current. It relates the magnetic field to the magnitude, direction, length, and proximity of the electric current. The Biot–Savart law is fundamental to magnetostatics, playing a role similar to that of Coulomb's law in electrostatics. When magnetostatics does not apply, the Biot–Savart law should be replaced by Jefimenko's equations. The law is valid in the magnetostatic approximation, and is consistent with both Ampère's circuital law and Gauss's law for magnetism. It is named after Jean-Baptiste Biot and Félix Savart, who discovered this relationship in 1820.

In physics, screening is the damping of electric fields caused by the presence of mobile charge carriers. It is an important part of the behavior of charge-carrying fluids, such as ionized gases, electrolytes, and charge carriers in electronic conductors . In a fluid, with a given permittivity ε, composed of electrically charged constituent particles, each pair of particles interact through the Coulomb force as

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in mechanical engineering and theoretical physics. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. It is a generalization of Laplace's equation, which is also frequently seen in physics. The equation is named after the French mathematician, geometer, and physicist Siméon Denis Poisson.

Electrostatics is a branch of physics that studies electric charges at rest.

Classical electromagnetism or classical electrodynamics is a branch of theoretical physics that studies the interactions between electric charges and currents using an extension of the classical Newtonian model. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics.

In electromagnetism, displacement current density is the quantity D/∂t appearing in Maxwell's equations that is defined in terms of the rate of change of D, the electric displacement field. Displacement current density has the same units as electric current density, and it is a source of the magnetic field just as actual current is. However it is not an electric current of moving charges, but a time-varying electric field. In physical materials, there is also a contribution from the slight motion of charges bound in atoms, called dielectric polarization.

Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is just the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequency do not change as a result of the scattering. This limit is valid as long as the photon energy is much smaller than the mass energy of the particle: , or equivalently, if the wavelength of the light is much greater than the Compton wavelength of the particle.

In physics, the Planck charge, denoted by , is one of the base units in the system of natural units called Planck units. It is a quantity of electric charge defined in terms of fundamental physical constants.

Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs units, or often just cgs units. The term "cgs units" is ambiguous and therefore to be avoided if possible: cgs contains within it several conflicting sets of electromagnetism units, not just Gaussian units, as described below.

Electrical work is the work done on a charged particle by an electric field. The equation for 'electrical' work is equivalent to that of 'mechanical' work:

Electric potential energy, or electrostatic potential energy, is a potential energy that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may have electric potential energy by virtue of two key elements: its own electric charge and its relative position to other electrically charged objects.

The Coulomb constant, the electric force constant, or the electrostatic constant (denoted ke, k or K) is a proportionality constant in electrodynamics equations. The value of this constant is dependent upon the medium that the charged objects are immersed in. In SI units, in the case of vacuum, it is equal to approximately 8987551787.3681764 N·m2·C−2 or 8.99×109 N·m2·C−2. It was named after the French physicist Charles-Augustin de Coulomb (1736–1806) who introduced Coulomb's law.

The physical constant ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has the CODATA value

The Larmor formula is used to calculate the total power radiated by a non relativistic point charge as it accelerates. This is used in the branch of physics known as electrodynamics and is not to be confused with the Larmor precession from classical nuclear magnetic resonance. It was first derived by J. J. Larmor in 1897, in the context of the wave theory of light.

A Gaussian surface is a closed surface in three-dimensional space through which the flux of a vector field is calculated; usually the gravitational field, the electric field, or magnetic field. It is an arbitrary closed surface S = ∂V used in conjunction with Gauss's law for the corresponding field by performing a surface integral, in order to calculate the total amount of the source quantity enclosed; e.g., amount of gravitational mass as the source of the gravitational field or amount of electric charge as the source of the electrostatic field, or vice versa: calculate the fields for the source distribution.

In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation play a central role in computational chemistry and physics for computing properties of molecules and aggregates of molecules, such as thermal conductivity, specific heat, electrical conductivity, optical, and magnetic properties, and reactivity.