Uniqueness theorem for the unbounded domain

Wc now examine the solution of Maxwell s equation in an unbounded domain with the given distribution of electromagnetic parameters e, //, and r. The electromagnetic field is generated by the sources (extraneous currents j ), concentrated within some local domain Q. Using the energy inequality (8.112), we can prove that there is only one (unique) solution of this problem. 

As in the case of the boundary-value problem considered above, wc a.ssumc first that there may be two different solutions of the Maxwell s equations in the unbounded domain with the same source j . Introducing the difference of those solutions, the electromagnetic field E, H, we notice immediately that the. source of this field is zero 

From this equation it follows immediately that in any lossy medium a 0), the electric field is equal to zero inside Og as well  

Formula (8.122) holds for any point r of the space, because for any r we can always select a radius R big enough that r EOg. Substituting (8.122) in the second Maxwell s equation, we conclude that magnetic field H is also identically equal to zero (assuming w 7 0). 

This completes the proof of the uniqueness theorem for the unbounded domain. 

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