# Coulomb’s Law

^{18 }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 Q_{1} and Q_{2} is:

a) Along the line joining them

b) Directly proportional to the product Q_{1}Q_{2} of the charges

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

Mathematically we can write as below.

F = kQ_{1}Q_{2}/ R^{2} ……………………(1)

where k is the proportionality constant. In SI units, charges Q_{1} and Q_{2}are in coulombs (C), the distance R is in meters (m), and the force F is in newtons (N) so that k = 1/4πξ_{0}. The constant so is known as the permittivity of free space (in farads per meter) and has the value

ξ_{0} = 8.854×10^{-12}= 10^{-9}/ 36π F/m

and k = 1/4πξ_{0} = 9×10^{9} m/F

Thus from equation (1), we can write as

F = (Q_{1}Q_{2}) / 4πξ_{0}R^{2}

If point charges Q_{1}and Q_{2} are located at points having position vectors **r _{1}** and

**r**, then the force

_{2}**F**on Q

_{12}_{2}due to Q

_{1}, shown in figure below, is given by

**F _{12}** = [(Q

_{1}Q

_{2}) / 4πξ

_{0}R

^{2}]

**a**…………………(2)

_{R12}Where **a _{R12} ** is unit vector in the direction of force experienced by charge Q

_{2}by Q

_{1}.

Here, **R _{12}** =

**r**–

_{2}**r**and R is the magnitude of vector

_{1}**R**.

_{12}Therefore, unit vector in the direction of **R _{12}**,

**a _{R12}** =

**R**/ R

_{12}Therefore from equation (2), force **F _{12}** on Q

_{2}due to Q

_{1}

**F _{12}** = [(Q

_{1}Q

_{2}) / 4πξ

_{0}R

^{2}]

**a**

_{R12}**⇒****F _{12}** = [(Q

_{1}Q

_{2}) / 4πξ

_{0}R

^{3}]

**R**

_{12} = [(Q_{1}Q_{2}) / 4πξ_{0}R^{3}] (**r _{2}** –

**r**)

_{1}**Some important points about Coulomb’s Law:**

a) It shall be noted that from the figure above that, the force **F _{21}**, on Q

_{1}due to Q

_{2}is given by,

**F _{21}** = F

_{12}

**a**= F

_{R21}_{12}(-

**a**)

_{R12}**⇒ ****F _{21}** = –

**F**

_{12}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 Q_{1} and Q_{2} must be large compared with the linear dimensions of the bodies; that is, Q_{1} and Q_{2} must be point charges.

c) Q_{1 }and Q_{2}must 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 Q_{1}, Q_{2},……,Q_{N} located, respectively, at points with position vectors **r _{1}**,

**r**,. . .,

_{2}**r**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 Q

_{N}_{1}, Q

_{2},. . ., Q

_{N}. Therefore,