[ 528 ] 



LXIII. Intelligence and Miscellaneous Articles. 



THE EFFECTS OF SELF-INDUCTION AND DISTRIBUTED STATIC 

 CAPACITY IN A CONDUCTOR. BY FREDERICK BEDELL, PH.D., 

 AND ALBERT C. CREHORE, PH.D. 



TTrlE solution obtained by Sir Wm. Thomson for the variation of 

 -*- the current and the potential at different points in a conductor 

 possessing static capacity is given By Mascart and Joubert, 

 VElectricite et Le Magnetisme,' vol. i. § 233, and is treated at 

 length by Mr. T. H. Blakesley in his book on Alternating Currents. 

 The object of the present cormnunicaiion is to give the solution 

 for the case of a conductor possessing self-induction as well as 

 distributed capacity, and to note the effects produced by the intro- 

 duction of the self-induction. 



The rate of change of the charge on an element of the cable is 

 equal to the difference of the currents flowing into and out from it ; 

 and so, writing q for charge and i for current at any time, and so 

 for the distance of any point from the origin — positive direction 

 being that of current-flow — we have 



dq di 



ctt dec 



If e is the potential of an element, its charge is q~Oedx, where 

 C denotes the capacity per unit length of the cable, and the first 

 equation may be written 



_ de di 



°s=-s a) 



By Ohm's law the current in an element is equal to the total 

 E.M.F. (the sum of the impressed and that of self-induction) 

 divided by the resistance ; and, if E is the resistance per unit 

 length and we assume the back E.M.F. of self-induction per unit 

 length to be equal to the rate of change of the current multiplied 

 by a constant L, we may write 



'= ss— < 2 > 



In some cases this assumption may approximately represent the 

 true effect of self-induction, and the results obtained from this 

 particular assumption may show the nature of the effect of self- 

 induction even in cases where the assumption is not justifiable. 



The differential equation for potential is obtained by eliminating 

 i from (1) and (2), and is 



The current equation, obtained in the same way, is similar when i 

 is written instead of e. 



