Treatment of Electrodynamics* 211 



are only concerned to find the simplest means of calculating 

 observed effects of electric currents and magnets. We will 

 therefore take the above rules and apply them to certain 

 cases. 



A conduction current consists in a continuous succession 

 of moving charges. Let there be a short element of 

 wire, of length dl. in which there are on the average 

 n' electrons per unit length, each of charge e, and travelling 

 with the average velocity V. Then neV is the total 

 charge conveyed per second across a section of the wire, 

 i. e. the current, in electrostatic measure. In electrodynamic 



ne^ 

 units the current is = i. 



c 



In the short length considered, however, there will be 



ndl electrons, so that the total electrodynamic effect of 



neVdl 



that length will be proportional to = idl. We may 



c 



accordingly write the former equations in a form suitable 



for conduction currents as : — 



u i\dh 



H =— ■■— 3 , 



X y.Z 



F x = + Ri 2 dl 2 sin 2 , F y = - H* 2 dl 2 cos 2 , 



where 6 2 * s ^ ne angle whose tangent is u 2 -r-W2« 



The resultant force is therefore ~F = lii 2 dl 2 . 



The equations may, of course, be transformed, without 

 difficulty, into the usual polar coordinates, or into the language 

 of vector algebra. 



It is now convenient to deal with the magnetic action of 

 a rectangular elementary circuit. Two principal cases arise 

 according as the field is required at a point lying in the axis 

 of the small circuit or in its plane. In the first case, let dy dz 

 be the area of the circuit, and the point considered be at a, 

 distance r along the axis of ,i\ Then one of the sides, of 

 length dy, situated ^dz from the axis of y, will produce a 



field — ~, of which the component in the axial direction 



is — ~ X — -. The transverse components of the opposite sides 



neutralize, leaving an axial resultant amounting to four 

 times the above, or 



7TT , * dy dz _ 2% dk. 

 ir '2 r r* 



Phil. Mag. Ser. 6. Yol. 17. No. 98. Feb. 1909. U 



