Steady Currents in Media Without Source of EMF

Consider the boundary between two media of different conductivities σ1 and σ2. Since charge can neither be piling up nor disappearing at a boundary, the current towards the boundary on one side must be equal to the away from it on the other side. Thus
            J₁ dS = j₂ . dS or j_1n =  j_2n,                            … (1)
Or the normal component of j must be continuous across the interface. This relation can also be written as
            σ1E₁n = σ2 E_2n .                                … (2)
So long as there are no sources of emf in either medium, Ï• E.dl = 0 for a closed path which links both media. Therefore
            E_1t = E_2t
Or            j_1t /σ1 = j_2t /σ2.
Combining Eqs. (1) and (4), we get
σ1 cot θ1 = σ2 cot θ2.
It is the equation for the law of refraction.
We have a situation similar that we found for dielectric. Thus we see that at the boundary between two media of different conductivities the lines of flow are refracted in a manner analogous to that of the lines of force at the boundary between two dielectrics.
This is an example of the similarity between conductors and dielectrics.


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