MATHEMATICAL TREATMENT OF COULOMB'S LAW

Consider two charges of magnitude 𝑞₁ and 𝑞₂ , assumed to be concentrated as point sources at a distance d from each other.

According to the second law of electrostatics, the force F between them may be expressed mathematically as:

F∝q1×qkmfafaie2

F∝1d2

or        F∝q1q2d2

F=k⋅q1q2d2N

                                                      ..................................(1)

where, N is Newton and k is a Constant depending on the nature of the medium between the charges. 

The value of k has been found experimentally, 

k=14πϵ0ϵr

                                                     ....................................(2)

where ,

ϵr

ϵ0

ϵ0

ϵ0

ϵ0

𝜖ᵣ is the relative permittivity of the medium intervening the charges

𝜖₀ is the permittivity of the evacuated free space

The value of 𝜖₀ in rationalised  M.K.S. unit is 8.854×10⁻¹² farads per meter.

When d is measured in meters, the constant 4𝜋 , the proportionality factor for the rationalised M.K.S. units.

Substituting the value of k from Eq. (2) in Eq. (1) we get,

F=q1q24πϵ0ϵrd2N

where,

𝑞₁ and 𝑞₂ are measured in coulombs

d in meters

𝜖 in farad / meter

F in newtons

For air         𝜖ᵣ = 1

∴    F=q1q24πϵ0d2N

when     

𝑞₁ = 𝑞₂ = 1 coulomb

d = 1 meter

𝜖₀ =  8.854×10⁻¹² farad / meter

F=1×14π×8.854×10−12×12

F=8.997×109