Coulomb's law, or Coulomb's inversesquare law, is a law of physics that describes force interacting between static electrically charged particles. In its scalar form, the law is:
where k_{e} is Coulomb's constant (k_{e} = ×10^{9} N m^{2} C^{−2}), q_{1} and q_{2} are the signed magnitudes of the charges, and the scalar r is the distance between the charges. The force of interaction between the charges is attractive if the charges have opposite signs (i.e., F is negative) and repulsive if likesigned (i.e., F is positive). 8.99
The law was first published in 1784 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. It is analogous to Isaac Newton's inversesquare law of universal gravitation. Coulomb's law can be used to derive Gauss's law, and vice versa. The law has been tested extensively, and all observations have upheld the law's principle.
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. 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.^{[1]}^{[2]} 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.^{[1]} 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.^{[3]} 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.^{[4]}
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^{[5]} and Alessandro Volta, both of whom measured the force between plates of a capacitor, and Franz Aepinus who supposed the inversesquare law in 1758.^{[6]}
Based on experiments with electrically charged spheres, Joseph Priestley of England was among the first to propose that electrical force followed an inversesquare law, similar to Newton's law of universal gravitation. However, he did not generalize or elaborate on this.^{[7]} In 1767, he conjectured that the force between charges varied as the inverse square of the distance.^{[8]}^{[9]}
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}.^{[10]}
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.^{[11]}
Finally, in 1785, the French physicist CharlesAugustin 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.^{[12]} 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 metalcoated 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 inversesquare proportionality law.
Coulomb's law states that:
The magnitude of the electrostatic force of attraction 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.^{[12]}
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
where k_{e} is Coulomb's constant (k_{e} = 5517873681764×10^{9} N m^{2} C^{−2}), 8.987q_{1} and q_{2} are the signed magnitudes of the charges, the scalar r is the distance between the charges, the vector r_{21} = r_{1} − r_{2} is the vectorial distance between the charges, and r̂_{21} = r_{21}/r_{21} (a unit vector pointing from q_{2} to q_{1}). The vector form of the equation calculates the force F_{1} applied on q_{1} by q_{2}. If r_{12} is used instead, then the effect on q_{2} can be found. It can be also calculated using Newton's third law: F_{2} = −F_{1}.
When the electromagnetic theory is expressed using the standard SI units, force is measured in newtons, charge in coulombs, and distance in metres. Coulomb's constant is given by k_{e} = ^{1}⁄_{4πε0}. The constant ε_{0} is the permittivity of free space in C^{2} m^{−2} N^{−1}. And ε is the relative permittivity of the material in which the charges are immersed, and is dimensionless.
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:
Cgs units are often preferred in the treatment of electromagnetism, as they greatly simplify formulas.^{[13]}
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 q_{t} depends on the electric field E that it finds itself in, such that F = q_{t}E. 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 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 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:
Coulomb's constant is a proportionality factor that appears in Coulomb's law as well as in other electricrelated formulas. Denoted k_{e}, it is also called the electric force constant or electrostatic constant, hence the subscript e.
The exact value of Coulomb's constant is:
There are three conditions to be fulfilled for the validity of Coulomb’s law:
When it is only 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 q_{1} and q_{2} as follows:
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 the two charges is repulsive; if the product is negative, the force between them is attractive.^{[14]}
Coulomb's law states that the electrostatic force F_{1} experienced by a charge, q_{1} at position r_{1}, in the vicinity of another charge, q_{2} at position r_{2}, in a vacuum is equal to:
where r_{21} = r_{1} − r_{2}, the unit vector r̂_{21} = r_{21}/r_{21}, and ε_{0} 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, r̂_{21}, parallel with the line from charge q_{2} to charge q_{1}.^{[15]} If both charges have the same sign (like charges) then the product q_{1}q_{2} is positive and the direction of the force on q_{1} is given by r̂_{21}; 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 given by −r̂_{21} = r̂_{12}; the charges attract each other.
The electrostatic force F_{2} experienced by q_{2}, according to Newton's third law, is F_{2} = −F_{1}.
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:
where q_{i} and r_{i} are the magnitude and position respectively of the ith charge, R̂_{i} is a unit vector in the direction of R_{i} = r − r_{i} (a vector pointing from charges q_{i} to q).^{[15]}
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,
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,
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,
The force on a small test charge q′ at position r in vacuum is given by the integral over the distribution of charge:
It is possible to verify Coulomb's law with a simple experiment. Consider two small spheres of mass m and samesign 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:


(1) 
and:


(2) 


(3) 
Let L_{1} be the distance between the charged spheres; the repulsion force between them F_{1}, assuming Coulomb's law is correct, is equal to


(Coulomb's law) 
so:


(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 L_{2} < L_{1} and the repulsion force between them will be:


(5) 
We know that F_{2} = mg tan θ_{2}. And:


(6) 
Measuring the angles θ_{1} and θ_{2} and the distance between the charges L_{1} and 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:


(7) 
Using this approximation, the relationship (6) becomes the much simpler expression:


(8) 
In this way, the verification is limited to measuring the distance between the charges and check that the division approximates the theoretical value.
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.
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.
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?
When making experiments with charged spheres of opposite charge the results were similar, as stated on page 73: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}.
Nonetheless, on page 74 the author infers that the actual action is related exactly to the inverse duplicate of the distance: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 ^{1}⁄_{6} 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.
On page 75, the authour compares the electric and gravitational forces: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.
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.
We may therefore conclude that the electric attraction and repulsion must be inversely as some power of the distance between that of the 2 + ^{1}⁄_{50} th and that of the 2 − ^{1}⁄_{50} th, and there is no reason to think that it differs at all from the inverse duplicate ratio.
Coulomb also showed that oppositely charged bodies obey an inversesquare law of attraction.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.
ke = H/m is not correct it must be F/m
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