Sioux12
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Hello 
As stated in this thread, the Uniqueness Theorem is very powerful for many applications. In Electromagnetism, the Theorem ensures that the solution of the Maxwell's equation (an electromagnetic field E, H) is unique, given certain conditions.
But I can't prove it when an infinite volume is to be considered.
In the frequency domain and in a finite volume V bounded by a surface S, given the sources (current densities and charge densities), the theorem is proved simply providing the boundary conditions: they are the exact values that the electric or the magnetic field tangential to S must have.
But if the volume V extends to infinite and so the surface S is infinite, the only conditions in this case are the radiation conditions, which don't provide a value for the electric or magnetic fields, but state only that they must be orthogonal on S and that the Poynting vector E x H must be outgoing from the surface. After the computations, in this file it is stated that "Infinite surfaces can be thought of as the limit of finite surfaces, so there is really no problem there if we specify the field vanishes (or is at least outward traveling) at ∞". But it is really a problem, because the difference fields (needed to prove the theorem) does not vanish!!
Do someone know this Theorem? Or know any book or any text which deals with it?
As stated in this thread, the Uniqueness Theorem is very powerful for many applications. In Electromagnetism, the Theorem ensures that the solution of the Maxwell's equation (an electromagnetic field E, H) is unique, given certain conditions.
But I can't prove it when an infinite volume is to be considered.
In the frequency domain and in a finite volume V bounded by a surface S, given the sources (current densities and charge densities), the theorem is proved simply providing the boundary conditions: they are the exact values that the electric or the magnetic field tangential to S must have.
But if the volume V extends to infinite and so the surface S is infinite, the only conditions in this case are the radiation conditions, which don't provide a value for the electric or magnetic fields, but state only that they must be orthogonal on S and that the Poynting vector E x H must be outgoing from the surface. After the computations, in this file it is stated that "Infinite surfaces can be thought of as the limit of finite surfaces, so there is really no problem there if we specify the field vanishes (or is at least outward traveling) at ∞". But it is really a problem, because the difference fields (needed to prove the theorem) does not vanish!!
Do someone know this Theorem? Or know any book or any text which deals with it?