﻿Capacity of a Pancake Coil. 731 



Simplifying Assumption as to Potential Distribution. 



An arithmetical computation of the e.m.f. induced in 

 various parts of the coil for a coil with a finite number of 

 turns revealed the fact that the e.m.f. induced between 

 a point on the surface of the coil and the centre is approxi- 

 mately proportional to the square of the distance of that 

 point from the centre. The computation above mentioned 

 consisted in calculating the e.m.f. induced between the 

 centre and a number of points at various distances from 

 the centre for the case of a coil having- a finite number 

 of equally spaced turns. Maxwell's formula in elliptic 

 integrals was used, and numerical results were tabulated. 

 These were then plotted, and the graph revealed the 

 approximate relation stated. 



The relation is frankly approximate, but is believed to be 

 accurate enough for the calculation of the coil capacity. 

 The computation which follows takes this for its starting- 

 point. 



General Plan of Attack. 



The first step will be to compute the distribution of charge 

 on the wires of the coil which will satisfy the law assumed 

 for the potential distribution. Then the quantity M(.i') will 

 be determined from the same law. The two expressions 

 will next be substituted in (1), and hence C will be 

 obtained. 



This w T ill be done for three cases — namely that of the 

 coil when ungrounded, and also when grounded — either at 

 the centre or else at the outer edge. 



The first part of the work consists, then, in the solution 

 of an electrostatic problem— namely that of finding the 

 charge distribution. The second part is ordinary inte- 

 gration. 



Solution of the Electrostatic Problem. 

 It is convenient to transform the cylindrical co-ordinates 



('S e, 0) 



to elliptical co-ordinates 



(«, v, 6) 



by the foi inula 

 where 



r-fjz = a cosh {u+jv), .... (2) 



or its equivalents r — a cosh u cos v. ) , 



. , . ► • • • • ('>) 



z = a sinh u sin v. j 



