no BELL SYSTEM TECHNICAL JOURNAL 



In order to calculate the field at some point A due to the distribution 

 (20), we must determine the retarded electric vector potential. Since 

 the integration is vectorial, it is convenient to deal with the cartesian 

 components of the magnetic current density 



M..=^^^, M..= -^^^- (24) 



P P 



The area of the element is p'dp'dip' so that the components of the 

 retarded potential are 



. ,''' r^-M^'e-'^-^-^' p'dp'dip' 

 rx — 





AA' 





AA' 



dp'dip', (25) 



F„=--l 1 -^, dpd^, ^^^^e=^ = -. 



Hence the components in the polar coordinates are 



dp'd<p', (26) 



F^ = — Fx sin (p -\- Fy cos (p 



b /»2xg-iMA' cos ((^ — <p') 



_ _p_ f"' r 



F, = 0. 



The distance AA' is 



^^' = \V2 - 2rp' cos I? + p'\ (27) 



where r is the distance OA, and d is the angle between OA and 0^'. 

 Since we are interested in the radiation field alone, we need retain only 

 those terms in (24) which vary inversely as the distance; the other 

 terms contribute nothing to the radiated power. Thus we let r 

 increase indefinitely, obtaining 



^^' = /- - p' cos d, (28) 



and then 



F^= - ^ — I e'^"' -=- ' cos {<p - ^')dp'd^'. (29) 



'*^'' Ja Jo 



If d and 6' are the angles made by OA and OA' with OZ, we have 



cos d = cos d cos 6' + sin 6 sin ^' cos (cp — <p') 



= sin ecos (^ - if'). (30) 



Since p' is small by comparison with the wave-length X, we can expand 

 the exponential term in the integrand into a power series and retain 



