Coulomb's Law

Coulomb's law is an experimental law formulated in 1785 by the French colonel, Charles Augustin de Coulomb. It deals with the force a point charge exerts on another point charge. By a point charge we mean a charge that is located on a body whose dimensions are much smaller than other relevant dimensions. For example, a collection of electric charges on a pinhead may be regarded as a point charge. Charges are generally measured in coulombs (C). One coulomb is approximately equivalent to 6 X 10¹⁸ electrons; it is a very large unit of charge because one electron charge e = -1.60 X 10^-19 C.

Coulomb's law states that the force between two point charges Q₁ and Q₂ is:

a) Along the line joining them

b) Directly proportional to the product Q₁Q₂ of the charges

c) Inversely proportional to the square of the distance R between them.

Mathematically we can write as below.

F = kQ₁Q₂/ R²  ��������(1)

where k is the proportionality constant. In SI units, charges Q₁ and Q₂are in coulombs (C), the distance R is in meters (m), and the force F is in newtons (N) so that k = 1/4p?₀. The constant so is known as the permittivity of free space (in farads per meter) and has the value

?₀ = 8.854x10^-12= 10^-9/ 36p F/m

and k = 1/4p?₀ = 9x10⁹ m/F

Thus from equation (1), we can write as

F = (Q₁Q₂) / 4p?₀R² 

If point charges Q₁and Q₂ are located at points having position vectors r₁ and r₂, then the force F₁₂on Q₂ due to Q₁, shown in figure below, is given by

F₁₂ = [(Q₁Q₂) / 4p?₀R²] a_R12 �������(2)

Where a_R12  is unit vector in the direction of force experienced by charge Q₂ by Q₁.

Here, R₁₂ = r₂� r₁ and R is the magnitude of vector R₁₂.

Therefore, unit vector in the direction of R₁₂,

a_R12 = R₁₂ / R

Therefore from equation (2), force F₁₂ on Q₂due to Q₁

F₁₂ = [(Q₁Q₂) / 4p?₀R²] a_R12 

?F₁₂ = [(Q₁Q₂) / 4p?₀R³]R₁₂

      = [(Q₁Q₂) / 4p?₀R³] (r₂ � r₁)

Some important points about Columb�s Law:

a) It shall be noted that from the figure above that, the force F₂₁, on Q₁ due to Q₂ is given         by,

F₂₁ = F₁₂a_R21 = F₁₂(-a_R12)

? F₂₁ = - F₁₂

Like charges (charges of the same sign) repel each other while unlike charges attract as shown in figure below.

b) The distance R between the charged bodies Q₁ and Q₂ must be large compared with the linear dimensions of the bodies; that is, Q₁ and Q₂ must be point charges.

c) Q₁ and Q₂must be static i.e. at rest.

d) If we have more than two point charges, we can use the principle of superposition to determine the force on a particular charge. The principle states that if there are N charges Q₁, Q₂,��,Q_N located, respectively, at points with position vectors r₁,  r₂,. . ., r_N then resultant force F on a charge Q located at point r is the vector sum of the forces exerted on Q by each of the charges Q₁, Q₂,. . ., Q_N.  Therefore,