# Couloub’s Law

As already mentioned above, the electric force between two particles

decreases with the inverse square of the distance, just as does the

gravitational force. Coulomb, who investigated the repulsion between

small balls charged by rubbing, discovered the dependence of the

electric force on distance through experiments. His experimental

results are summarized in coulomb’s Law -

The magnitude of the electric force that a particle exerts on

another particle is directly proportional to the product of their

charges and is inversely proportional to the square of the distance

between them. The direction of the force is along the line from one

particle to the other.

To express this law mathematically, we denote the charges on the

particles by q and q′ and distance by r. Coulomb’s Law is then

represented by the formula

F = [constant] x q q′ / r2.............(1)

This formula gives not only the magnitude of the force, but also

direction, if we interpret a positive value of the force F as repulsive and

a negative value as attractive. For instance, for the case of the force

exerted by a proton on an electron, the charges are q′ = e and q = -e,

and the formula (1) yields

F = [constant] x e x (-e) /r2 ..................(2)

which is negative, indicating attraction.

The electric force that the particle of charge q exerts on the particle of

charge q′ has the same magnitude as the force exerted by q′ on q, but

the opposite direction. These mutual forces are an action-reaction pair.

In SI units, the constant of proportionality in Coulomb’s Law the value

[constant] = 8.99 ×109 N.m2/c2 .................. (3)

This constant is traditionally written in the form

[constant] = 1/ 4 πε0 ............. (4)

ε0 = 8.85 × 10-12C2/(N.m2) ....................... (5)

The quantity ε0 (epsilon nought ) is called the permittivity constant. In

terms of the permittivity constant, Coulomb’s Law for the force that a

particle of charge q′ exerts on a particle of charge q becomes

Although the second expression on the right side of Eq. (6) is most

convenient for numerical calculations of the Coulomb force, the first

expression with 1/4π ε0 is generally used in manipulations involving

formulas. Of course, the two expressions are mathematically

equivalent, and they give the same results.

Coulomb’s Law applies to particles - electrons and protons - and also

to any small charged bodies, provided that the sizes of these bodies are

much smaller than the distance between them; such bodies are called

point charges, Equation (6) clearly resembles Newton’s law for the

gravitational force, F = GMm/r2. The constant 1/4π ε0 is analogous to

the gravitational constant G, and the electric charges are analogous to

the gravitating masses.

In SI the Coulomb is defined in terms of a standard electric current:

one Coulomb is the amount of electric charge that a current of one

ampere delivers in one second.