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Duality between j and D

J and D both are linearly related to the electric field E in many materials. Hence dual relationship between j and D exists. In the region, where non-conservative fields are absent, we have
        Conducting media                    Dielectric media
        J = σE                            D = ε E
        ∇.j = 0                            ∇.D = 0
        ∇ x j = 0                        ∇ x D = 0        … (1)

In a homogeneous material where ε and σ are constant, we have
        ∇ x j = 0                        ∇ x D = 0
and hence    j = - ∇Φ                            D = - ∇Φ,        … (2)

Here Φ, a potential function, satisfies function, satisfies Laplace’s equation.
Above equations show that any solution of j can be transformed into a solution for D and vice versa, by interchanging as
        J ↔ D and σ ↔ ε.
This is also true as we have similar type of relations for the boundary of two media.
        Conducting media                    Dielectric media
        Jn1 = jn2                            Dn1 = Dn2
        Jt11 = jt22                        Dt11 = Dt22.        … (3)

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