388 Lord Rayleigh on the Self-induction 



in which B/ represents the effective resistance and 1/ the 

 effective self-ind action. 



If the rate of alternation be very slow, so that p is small, 

 these equations give (as was to be expected) , R/ = R, and 



U=l{A+iri, (21) 



representing the self-induction for steady or slowly alternating- 

 currents. If we include the next terms, we see that as the 

 frequency increases, the self-induction begins to diminish. 

 At the same time the resistance begins to increase. 



These results are merely very special cases of a general 

 law*, from which we may learn that as the frequency of alter- 

 nation gradually increases fron zero to infinity, there is a 

 steady rise of resistance and accompanying fall of self-induc- 

 tion. The application of the general idea to the present case 

 is very simple. At slow rates of alternation the distribution 

 of current, being such as to make the resistance a minimum, 

 is uniform over the section; and this distribution, since it 

 involves magnetization of the outer parts of the cylinder, leads 

 to considerable self-induction, especially in iron. On the 

 other hand, when the rate of alternation is very rapid, the 

 endeavour is to make the self-induction a minimum irrespec- 

 tive of resistance. This object is attained by concentration 

 of the current into the outer layers. The magnetization of 

 the conductor is thus more and more avoided, but of course 

 at the expense of increased resistance. We may gather from 

 the general argument, what (19) and (20) in their actual 

 forms do not tell us, that as p increases without limit, R/ also 

 becomes infinite, while the part of 1/ depending upon the 

 magnetization of the conductor tends to zero. 



The increase of resistance proper (not merely of the 

 " throttling " due to the combined effect of resistance and 

 self-induction) in iron wires of moderate diameter subjected 

 to varying currents, is one of the most striking of Prof. 

 Hughes's results. So far as I am aware, neither Maxwell nor 

 any other theorist had anticipated that the alteration of resist- 

 ance would be important under such circumstances")*. 



In order to see under what conditions the alteration of 

 resistance (and of self-induction) would become sensible, we 



have to examine the value of =-5 "^w- - We w ^ * a ^ e ^ rs ^ 



* See preceding article. 



t In the paper referred to I have quoted Maxwell's calculation of 

 increased resistance and diminished self-induction due to the operation 

 of currents in a secondary circuit. 



