96 BELL SYSTEM TECHNICAL JOURNAL 



very thin filament, the moment is the product of the current and the 

 length of the element. 



It is the moment of the current element that determines its electro- 

 magnetic field. If the medium is non-dissipative, the actual expres- 

 sions for the field components are obtained in terms of an auxiliary 

 function called by Lorentz the retarded magnetic vector potential. For 

 an electric current element of moment p{t) this vector potential at any 

 point P is parallel to the current density and is a function of the 

 distance r from the element to P 



p{'-\) 



^ =— 1 —• (4) 



The quantity c has the dimensions of a velocity and it appears that the 

 action of the source travels outward with this velocity. But there is 

 another solution of (3) 



^(' + ^) 



One might wonder if this solution appertains in any way to the source; 

 that is not the case, however. If the moment p{t) is identically zero 

 prior to some instant / = /o, the field which can legitimately be attrib- 

 uted to the action of this source is also identically zero for any instant 

 / < /o- But (5) implies a non-vanishing field at distant points; it is as 

 if the effect appeared before the cause. Any other solution is a 

 combination of (4) and (5) and has to be rejected on the same grounds. 

 In terms of this auxiliary vector potential the field components can 

 be expressed as follows 



H = curM, ^ = - curl //, E = curl II dt. (6) 



at t e / 



•-'—00 



If the moment is harmonic of frequency/, we regard it as the real 

 part of /)e'"'. Then the vector potential and the field components are 

 the real parts of the following expressions 



A = —, , H = curl A, E = - itofuA -f ^ -. (7) 



47rr tue ^ ■^ 



where the phase constant 



X is the wave-length, and the time factor e'"' is implied. 



CO 2irf 2ir 



