ELECTROMAGNETIC THEORY 



Let us now examine the field of the currents and charges by aid of 

 the formulas 



P 

 E ^ — grad ^ — — A, 



H = curl A. 



Performing the indicated operations, 



curl A^ - ^ e-^^^^^^ln-u-}l^~ + ^ ^J dv, 



grad«l>= - j e-^r>ic)rp.j^(^^^j^t^\^dv, 



where n is a unit vector, parallel to r, drawn through the contributing 

 element. 



We see from these formulas that the magnetic field due to the 

 currents, and the electric field due to the charges, consist each of two 

 components; one varying inversely as the square of the distance from 

 the contributing element and the other inversely as the distance. 

 Writing p = ico = i-lirf, the orders of magnitude of the two com- 

 ponents are 1/r- and o^jc^ and their ratio is 2x(r/X), where X is the wave 

 length. 



The first component is the induction field, and involves the fre- 

 quency only through the exponential term ; the second is the radiation 

 field and involves the frequency linearly. 



If we are considering points in the system itself, and if the dimen- 

 sions of the system are so small that 27r(r/X) is small compared with 

 unity, the expressions reduce to 



curl A= - j -^^^dv, 



grad ^ = - j n^^ 



dv. 



If therefore the dimensions of the system are sufficiently small with 

 respect to the wave length, these expressions can be employed in cal- 

 culating the distribution of the currents and charges in the system. 

 This is usually the case in circuit theory, even at radio frequencies. 



At a great distance from the system, however, the case is quite 

 different. For no matter how large the wave length, X, if we consider 

 points outside the system such that 27r(r/X) is everywhere large com- 

 pared with unity, the second or radiation field will predominate. 

 This leads to the important conclusion that the field which determines 

 the distribution of currents and charges in the system is quite different 



