# 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.

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