MUTUAL IMPEDANCES OF PARALLEL WIRES 533 



system of charges and currents; r is the distance between dv and P; 

 and c is the velocity of Hght in free space. q{t — r/c) and u{t — r/c) 

 are the charge-density and the vector current-density, respectively, 

 in dv, not at the time / but at the slightly earlier time / — r/c, allowance 

 thus being made for the time of propagation of the effect from dv to P. 

 Thus in (2) the integration, made at the time t, which is that observed 

 at P, must include every volume-element dv which contained any 

 charges at the time / — r/c, whatever the motions of those charges; and 

 in (3) the integration must include every volume-element dv which 

 contained any current (moving charges) at the time / — r/c; moreover, 

 associated with each volume-element dv is a corresponding value of r. 



r denoting distance, ^ and A are called "potentials" because of 

 their inverse dependence on r and their direct dependence on the charge- 

 density q and the current-density u respectively. ^ is called the 

 "scalar potential" because it does not have direction in space; A 

 the "vector potential" because it has direction. These potentials 

 are qualified as being "retarded" potentials'^ because the values to be 

 taken for the charge-elements and current-elements are not their 

 actual values at the contemplated instant / but their "retarded" 

 values, that is, their values at the earlier instants i — r/c. (It is to 

 be remembered that the time t is that observed at the point P where 

 ^ and A are to be calculated.) 



In the way of a summary statement regarding the set of formulas 

 (2) to (7), we may say that electric charges, whether stationary or 

 moving, produce a scalar potential ^ calculable from (2), and thence 

 an electric field intensity Eq calculable from (4) ; and that if the charges 

 are in motion, thus constituting currents, they produce also a vector 

 potential A calculable from (3), and thence an additional electric 

 field intensity Eu calculable from (5) and a magnetic field intensity 

 H calculable from (6). Thus the total, or resultant, electric field 

 intensity E = Eq -\- Eu'is calculable from 



E= - grad ^ - ^ ^ • (8) 



If the contemplated point P for which E is calculated is in a conductor, 

 of resistivity p, where the current-density is w', there exists the addi- 

 tional relation E = pu' , in accordance with Ohm's law. 



Of the important contrasting principles enunciated in the section 

 entitled "The Two Parts of a Voltage, and Their Resultant," principle 

 " 1 " is an immediate consequence of equation (4) of this appendix, and 

 "2" is a consequence of (5) and (6) together. 



1" Sometimes called "propagated" potentials. 



