Equation of Electrical Flow. 423 



first term heats the conductor and the second term gives 

 energy to space. 



We may go somewhat further into the causes of such an in- 

 duced electromotive- force component if we 

 employ the geometrical mode of symboli- 

 zing the electric quantities. BC, the induced 

 electromotive-force line, should be at right 

 angles to the induction through the circuit, 

 for it is the rate of increase of the latter 

 which produces the former. Hence if A E 

 is a perpendicular let fall upon B C pro- 

 duced, A E will represent the phase of the 

 magnetic induction. But AC being in 

 phase with the current is in phase with the 

 field. Hence EAC, or CBD which is 

 equal to it, is a magnetic phase-lag, and A E may be said to 

 be in phase with the effective field, and therefore with the 

 induction. This suggests that if we employ the lower lines of 

 the figure to represent fields, we may make up a triangle 

 ACE such that A C is the impressed field, 

 C E an induced field, and A E an effective 

 field, of course when, as usual, projected on 

 a fixed line ; C E being perhaps, though by 

 no means certainly, at right angles to AE. 

 However, whether C E here has in any 

 case two components perpendicular and 

 parallel respectively to A E or not, it appears very certain that 

 the perpendicular component must exist. Assuming at first 

 that it alone exists, — 



J f we employ small letters : — 



v for impressed field =AC, 



/for induced field = CE, 



e for effective field =AE, 



I = coefficient of magnetic self-induction, so that 



/_ l dt ' 



and ix for the permeability, I for the rate of magnetic induction, 

 i. e. per square centim., we have 



7 dl I 



at fM 



To obtain an equation of energy from this we must multiply 

 (not by I, as analogy would at first sight perhaps dictate) by 



j- .dt x cross section, for the formula for energy is 



