Coulomb's law or Coulomb's inversesquare law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was studied and first published in 1783 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism. Nevertheless, the dependence of the electric force with distance had been proposed previously by Joseph Priestley^{[1]} and the dependence with both distance and charge had been discovered, but not published, by Henry Cavendish, prior to Coulomb's works.
Basic equation
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."
The scalar form of Coulomb's law is an expression for the magnitude and sign of the electrostatic force between two idealized point charges, small in size compared to their separation. This force (F) acting simultaneously on point charges (q_{1}) and (q_{2}), is given by
In the more useful vectorform statement, the force in the equation is a vector force acting on either point charge, so directed as to push it away from the other point charge; the righthand side of the equation, in this case, must have an additional product term of a unit vector pointing in one of two opposite directions, e.g., from q_{1} to q_{2} if the force is acting on q_{2}; the charges may have either sign and the sign of their product determines the ultimate direction of that force. Thus, the vector force pushing the charges away from each other (pulling towards each other if negative) is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The square of the distance part arises from the fact that the force field due to an isolated point charge is uniform in all directions and gets "diluted" with distance as much as the area of a sphere centered on the point charge expands with its radius.
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 vector happens to be parallel to the electric field vector at that point, with that point charge (or "test charge") removed.
Coulomb's law can also be interpreted in terms of atomic units with the force expressed in Hartrees per Bohr radius, the charge in terms of the elementary charge, and the distances in terms of the Bohr radius.
Electric field
Vector form
In order to obtain both the magnitude and direction of the force on a charge, q_{1} at position , experiencing a field due to the presence of another charge, q_{2} at position , the full vector form of Coulomb's law is required.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 ; 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 ; the charges attract each other.
System of discrete charges
The principle of linear superposition may be used to calculate the force on a small test charge, q, due to a system of N discrete charges: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 gives the charge per unit length at position , and is an infinitesimal element of length,
 .^{[9]}
 ^{[8]}
Graphical representation
Below is a graphical representation of Coulomb's law, when q_{1}q_{2} > 0. The vector is the force experienced by q_{1}. The vector is the force experienced by q_{2}. Their magnitudes will always be equal. The vector is the displacement vector between two charges (q_{1} and q_{2}).Electrostatic approximation
In either formulation, Coulomb’s law is fully accurate only when the objects are stationary, and remains approximately correct only for slow movement. These conditions are collectively known as the electrostatic approximation. When movement takes place, magnetic fields are produced which alter the force on the two objects. The magnetic interaction between moving charges may be thought of as a manifestation of the force from the electrostatic field but with Einstein’s theory of relativity taken into consideration.Table of derived quantities
Particle property  Relationship  Field property  
Vector quantity 

 
Relationship  
Scalar quantity 


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