Inductance of a Conductor due to External Flux

Now we shall estimate the flux linkages of the conductor due to the external flux. For this we will consider the flux linkages of an isolated conductor due to that portion of the external flux which lies between two points distant and meters from centre of conductor P₁ and P₂ are two such points as shown in the Fig. 1.

Fig. 1

     
The conductor shown in the Fig. 1 carries current I. The flux paths are concentric circles around the conductor between P₁ and P₂.

       Consider a tubular element which is x meters from centre of conductor. The field intensity at this point is  H_x. The mmf around the element is

       The flux density B_x at this point is given by

       The flux dΦ in the tubular element is given by,
                      dΦ = B_x x dx x 1m (Axial length is considered as 1m)

       The flux linkages dψ per meter equal to dΦ since flux external to the conductor links all the current in the conductor. The total flux linkage between P₁ and P₂ are obtained by integrating dψ from D₁ to D₂ 

       Since the    μ = μ_o μ_r
       For Relatively permeability, μ_r = 1

       The inductance due to flux included between P₁ and P₂ only is,

       In the external flux is considered to be extended from the surface of conductor to infinity then total flux linkages is given by,

       Overall flux linkages is given by