FORCE BETWEEN MAGNETIC POLES

The force of attraction or repulsion between unlike or like magnetic poles, was first measured experimentally by Coulomb in 1820. With the help of torsion balance Coulomb found that "The force of attraction or repulsion between isolated point magnetic poles was proportional to the pole strength (m) and inversely proportional to the square of the distance between the poles". Although it is impossible to get isolated point magnetic poles in practice, but the poles of a long thin magnet may be assumed as point poles.

Consider two isolated magnetic poles of strength m1 and m2 placed at a distance r, in air. 

The force F between the poles is given by:

F∝m1×m2

F∝1r2


∵   F∝m1×m2r2

                                  .................................................(i)

If the medium in between the poles is some other media, other than air, the force F is given by the expression:

F∝m1×m2μr⋅r2

                                  .................................................(ii)

where 𝜇ᵣ is known as Relative Permeability of that medium. Its magnitude for air and non-magnetic materials is unity.

In general, the  equation (ii) can be written as

F=k⋅m1×m2μr⋅r2

                                 .................................................(iii) 

where k is a Constant of Proportionality depending upon the units of measurement.

In rationalized M.K.S. units the value of k is given by:

k =  107(4π)2   or   10716π2

Since F is measured in newtons, r in meters, then relation (iii) reduces to

F= m1×m2×10716π2×μr⋅r2  N   (when N is newton)

                =m1×m24π(4π×10−7)μr⋅r2   N


=m1×m24π⋅μ0⋅μr⋅r2   N



where  𝜇₀ = 4𝜋 ⨯ 10 ⁻⁷ is known as the Permeability of Free Space.

Since  𝜇ᵣ = 1 for air or non-magnetic materials, the force for air,

F=m1×m24πμ0⋅r2  N