﻿Potentials of Moving Charges. 379' 



(4) 



Lei us suppose a straight linear uniform current to 

 arise from continuous and uniform rush of electrons in the 

 conducting wire in the direction of the current. Viewed 

 from a system of axes moving with the common velocity of 

 the electrons the phenomena reduce, as far as the rushing 

 electrons are concerned, to the case of a linear and uniform 

 distribution of electric charge. If N electrons each with 

 charge — e be supposed to rush with velocity v to the 

 observer in the rest-system the linear density of static charge 

 is — N*. 



From the ordinary theory of potential, the potential <E>' for 

 such a distribution is — 2N<? logV where r l2 =y' 2 + z' 2 and 

 (F', G', H') = 0. Transforming to a moving system according 

 to our formula? we should have 



F = Kv&=-2/eNev\ogr, G=Gr'=0, H = H' = 0, 



<5? = K & = -2/cNelogr [v r-=r']. 



The magnetic field therefore would be given by h x = 0, 



2xNevz 1 2/eNm/ , , . . „ , 

 n v = 5 — , h = ^ —■ and the electric held would be 



, , A 7 2/cN^y , 2/cNez T ,, 

 given by d x — 0, Oy^= y^ , a*= ^— . In the con- 

 ducting wire, however, there is also a linear distribution of 

 positive nuclei at rest of which the potential would be 

 + 2N<? log r. 



The electric field due to these would be given by d/ = 0, 

 , mey ■ , M 2-Nez 



The resultant electric field would therefore have the 

 components 0, !L ~V (1— /e), - — ~ (1 — k), and is of the order 



v 2 

 of 2 . The magnetic field is of finite magnitude and cir- 

 cular round the wire, the resultant being which is 



., . , . , . . 2 (current) .„ 



quite in accord with the expression it we put 



k±s ev iS ev 

 current = or neglecting quantities of the order 



v 2 

 2 in comparison with unity. 



