62 /,. Page — Fundamental Relations of Electrodynamics. 



The force E, as already noted, lias the direction of the line 

 of force through P, as observed in K (0) ; that is, the direction 

 OP, where O is the apparent position of the charge to an 

 observer in K (0) at the instant considered. 



Thus, by means of the principle of relativity we have been 

 able to derive from the laws of electrostatics, with considerable 

 ease, an expression which Heaviside has derived from the 

 electromagnetic equations only by the use of somewhat compli- 

 cated mathematical processes. 



The relations between the components of E at P and E' 

 at P' follow at once from the expressions we have already 

 derived. 



E. = E '' 



E 



c 



E v = 



e z = e; 



Force between Current Elements. 



We can consider a current as made np of a given quantity of 

 positive electricity moving with a given drift velocity along the 

 wire in the direction of the current, and some other given 

 quantity of negative electricity moving with some other given 

 drift velocity in the direction opposite to that of the current. 

 Let -Uj be the velocity of the positive electricity, and w a that of 

 the negative electricity. Let A.,, be the linear density, or the 

 quantity of moving positive electricity per unit length of wire, 

 and A a the quantity of moving negative electricity per unit 

 length of wire. Consider an element of the wire of length ds. 

 Then we can define a current element as (\,i*j + \ a w a ) ds. Now 

 this element of wire is as a whole uncharged. So there must 

 be a quantity of positive electricity (k — \) ds, and a quantity 

 of negative electricity (k — X a ) ds at rest in the element, k being 

 some constant. As the current is due to that part of the 

 charge in the wire which is in motion, our problem reduces to 

 a consideration of the forces between two charges both of 

 which are moving relative to the observer. 



In order to make our reasoning as simple as possible, we 

 shall confine ourselves to currents lying in the same plane. 

 There is no difficulty in extending the reasoning to currents 

 which do not lie in the same plane, but in that case the demon- 

 stration becomes a little more complicated. 



