Forms of the reciprocity theorems are used in many electromagnetic applications, such as analyzing electrical networks and eigen-operators ).
12.
Now for a given disposition of the antennas, the reciprocity theorem requires that the power transfer is equally effective in each direction, i . e.
13.
The situation changed when Gauss used Gauss sums to show that quadratic fields are subfields of cyclotomic fields, and implicitly deduced quadratic reciprocity from a reciprocity theorem for cyclotomic fields.
14.
This idea of using roots of unity was introduced by Chebotarev in his proof of Chebotarev's density theorem, and used shortly afterwards by Artin to prove his reciprocity theorem.
15.
Each small voltage induces a current and these currents circulate through a small part of the antenna reciprocity theorem, it is possible to prove that the Th�venin equivalent circuit of a receiving antenna is:
16.
As a result of the Reciprocity Theorem the same loop will act as a receiver of high frequency noise and this will be coupled back into the signal circuits, with the potential to cause serious signal corruption and data loss.
17.
Another special case of the Lorentz reciprocity theorem applies when the volume " V " entirely contains " both " of the localized sources ( or alternatively if " V " intersects " neither " of the sources ).
18.
This result ( along with the following simplifications ) is sometimes called the "'Rayleigh-Carson reciprocity theorem "', after Lord Rayleigh's work on sound waves and an extension by John R . Carson ( 1924; 1930 ) to applications for radio frequency antennas.
19.
They need " not " be real & mdash; complex values correspond to materials with losses, such as conductors with finite conductivity ? ( which is included in ? via \ varepsilon \ rightarrow \ varepsilon + i \ sigma / \ omega ) & mdash; and because of this the reciprocity theorem does " not " require time reversal invariance.
20.
Since the left-hand side of the Lorentz reciprocity theorem vanishes for integration over all space with any non-zero losses, it must also vanish in the limit as the losses go to zero . ( Note that we implicitly assumed the standard boundary condition of zero incoming waves from infinity, because otherwise even an infinitesimal loss would eliminate the incoming waves and the limit would not give the lossless solution .)
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