231] ELECTROMAGNETISM. 



For a circular cross-section of radius a 



ra ft i ^/ a a _ 2 z _______ 



(9) Z = 2n> log - - ===== cfc = 47m> (E - V^ 2 - a 2 ). 



J Ct Ji V ft - ^ 



If in equation (2) we insert the number of turns of wire per unit 

 of length of the line of force, m, since n = Zirpm, 



(10) . H=4f7rml, 



or the force depends only on the amount of current per unit of 

 length. In case the radius of the tore is increased indefinitely, 

 so that we get an infinitely long straight coil, m is the number of 

 turns per unit of length of the coil, and we have within a uniform 

 field of the magnitude 47rm/. If any coil of n' turns be wound 

 on outside, the mutual inductance will be 



It is noticeable in all these cases that it is of no importance whether 

 the outer coil is in contact with the inner or not, for in any case 

 it is threaded by the whole flux of force. If there were any field 

 external to the tore, the case would be different. It is however 

 necessary that the tore be entirely filled by the medium of induc- 

 tivity p. The formulae of this section are applicable to induction 

 coils and transformers, providing the coils are endless. The line- 

 integral of magnetic force 4>7rnl is called the magnetomotive force, 

 and the problem of finding the magnetic induction in the tore is 

 the same as that of finding the current in a tore of conductivity 

 IJL in which there is an impressed electromotive force of the 

 amount 4-Tm/, the lines of flow being circles. In case the cross 

 section of the tore is small compared to its radius, we may neglect 

 the curvature of the coil, and find the reluctance ( 184), by 174, 

 so that we have 



/ \ T j 4.- TT-I Magnetomotive force ^irnl 

 ( 1 1 ) Induction Flux = & _. . - = = . 



Reluctance I 



?8 



This formula is used in practice in finding the flux in the field 

 magnet of a dynamo-electric machine, although it is accurate only 

 in the case that we have treated, where all the tubes of force are 

 encircled by all the current turns, so that the numerator is the 

 same for every tube. Any tube being partly in iron and partly 

 in air, the reluctance of any infinitesimal tube is found by the 

 formula for the resistance of conductors in series, as 



