- #1

Unconscious

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__Boundary conditions__

If there are superficial currents (electric and magnetic) impressed on the boundary between two media, we have these discontinuities for the tangential components of the fields:

$$\mathbf{n}\times(\mathbf{H}^+-\mathbf{H}^-)=\mathbf{J}_S$$

$$\mathbf{n}\times(\mathbf{E}^+-\mathbf{E}^-)=-\mathbf{J}_{m_S}$$

Consequently in these cases I never know what the field is in the points that constitute the boundary.

__Uniqueness theorem__

Given a region of volume τ occupied by a linear, stationary, dissipative for conductivity and non-dispersive medium, bounded by a closed region S, the field is unique in every point of τ that meets the following conditions:

1. both solution of Maxwell's equations and constitutive relations;

2. at least one of the two tangential components of the electric or magnetic field is assigned to each point of S.

__Love theorem (or equivalence theorem)__

Given a field in a region of space V, solution of Maxwell's equations, it is possible to express the field inside a volume τ⊂V like that generated by fictitious surface current sources present on the surface S that surrounds τ. These sources on S are obtained from the knowledge of the field in V, in particular from the tangential components of electric and magnetic fields on S:

$$\mathbf{n}\times(\mathbf{0}-\mathbf{H}^-)=\mathbf{J}_S$$

$$\mathbf{n}\times(\mathbf{0}-\mathbf{E}^-)=-\mathbf{J}_{m_S}$$

where the normal is outgoing from S, and the only fields that appear (superscript '-') are those inside τ in the vicinity of S (they are obtained as a limit approaching a point of S from the inside of τ).

__Personal doubt__

The proof of Love's theorem, at least as an idea, is simple having in mind the uniqueness theorem. However I see a problem of principle. If instead of using these fictitious sources, I directly imposed the tangential components of the fields on S, then the Love theorem would be very clear to me.

Going through the use of equivalent surface currents, I no longer return because in this way I have not imposed the value of the tangential field on S, but I have only imposed a discontinuity between the tangential fields inside S and outside S But the field exactly on S? I do not know its values in this way, for what was said at the beginning (boundary conditions). Consequently the uniqueness theorem for the field inside τ can no longer be applied correctly, because hypothesis 2 fails.

Do you agree or am I missing something?