Electromagnetic irradiation

1) the function to climb of Green:

In electromagnetism a impulsive answer in the space in how much is considered the sources has a sinusoidale course in the time but they are null ovunque unless in the point r' in which they are locate, it concurs to determine the produced electromagnetic largenesses from the source in the points r of the space, in particular the impulsive answer can be determined upgrades them carrier in this case coincides with the function to climb of Green, or pu² to determine the field where stavolta the impulsive answer is the diadica function of Green which is more complex dealing itself than a carrier. In order to determine the function to climb of Green reference is made the equation of the waves to climb (…projected on one of the cartesian aces) in presence of space impulsive currents impresse of which the function to climb of Green is solution.

The origin of the system of coordinates is the point where the source is placed, if we consider the equation previous in the external points to the sphere that comprises sources has from which dividend for r and taking the solution rG is had whose solution is the sum of a progressive wave and a regressive wave, this last one does not come considered in how much physically does not have sense a wave that collassa. Dividend for r finds the expression of the function to climb of Green where coefficient C is found integrating the equation on the spherical volume of containing beamr _{the 0} sources, using the spherical coordinates and dividend for r² is obtained that, carrying out the calculations door to that for r®0 becomes infinite but considering the infinites of advanced order and becoming simpler is obtained and therefore . After all whichever of the space upgrades them carrier of a point is found with la extended to the single volume that contains sources. In particular in the case of a impulsive source is found whose rotor in spherical coordinates, become simpler holding account that for via of the axial symmetry of the source, gives

and is therefore always orthogonal to the source, replacing it in the obtains therefore lies in a meridian that contains the direction of sources. From the previous ones making the the induction field is obtained while col obtains the cancellation field.

 

2) Irradiation from sources of ended dimensions:

A source not more impulsive is considered but than ended dimensions, a its point is taken like origin of the system of coordinates, all the other points of the source to be distant r' from it and R from the observation point p that instead tos be distant r from the origin. The H field is gained from upgrades them carrier To which the function is legacy to sources through the impulsive answer…(to climb of Green)

, clearly for r®0 of the terms within parenthesis considers alone while for r®¥ it is considered solo jb . The integral that obtains for the field to great distance becomes simpler in how much if the maximum D dimension of the source is much minor of the R distance (…between a point of the source and the point p) is had that the payer R₀ @ r₀ and therefore is constant in the V' volume that encloses the source and can be carried outside from the integral, like pure to the R denominator @ r while in the argument of the exponential is replaced where is the payer in the direction that joins the origin with the source point p' that it tos be distant r', after all ha replacing itself in the is obtained .

The carateristiche of the obtained fields are described from the cancellation condition:

.

 

3) electromagnetic Reciprocity:

The field produced from a monochromatic sourceTo ₁ constituted from magnetic currents impresse J_m1 and J electrical workers_i1 satisfies the equations , , while for the analogous ₂ sourceTo operating to the same frequency has , , multiplying respective for H₂ , and₂ , - H₁ , - and₁ and adding member to member obtains to 1° the member one amount that with the solità vectorial identity it comes brought back to . Integrating to the 1° and 2° the member on one volume V that contains the sources, to 1° the member can after all be applied to the theorem of the divergence obtaining the theorem of valid reciprocity for means isotropo and be delineated where account has been kept that the J sources_i1 and J_m1 is not null single in volume V₁ that the source To ₁ contains and analogous for the source To₂. From the cancellation conditions one deduces that the field to the infinite becomes infinitesimal therefore integrating on one volume V that contains all the space cancels the flow through the S surface that encloses it and the equality of the reactions is had therefore from which following can be deduced turn out to you:

to)       considering absent the magnetic currents impresse in two thin dipoles that can be contained in two cylinders and decomposing the integral of volume in the series of an integral of surface that for the presence of J the current characterizes and of an integral of line that, for the presence of and, characterizes tension V, dividend for I₁I₂ obtains the equality of the mutual stiffnesses.

b)       Applying the equality of the reactions to a source To and a impulsive source of test space having is had that concurs to gain the field produced from the source To in a point r.

 

4) Theorem of equivalence:

It is supposed of having a source To and a source of unitary test that concurs to estimate the reaction, for the equality of the reactions it is also equal to_{the tA} , the theorem of reciprocity on a volume is considered but that contains all the space ad.eccezione.del volume that contains the source To and is enclosed from the S surface_To , in such a way does not have the integral on the volume enclosed from S_To , obtains from which through the vectorial relation dove is obtained electrical worker is the superficial current while is magnetic the superficial current.

He has himself therefore that the field and nella ₀ direction oft product gives it source To is the same product gives them to superficial currents equivalents that find on a surface that it encloses the source To.