394, 395] Passage of Electricity through Dielectrics 347 



and integrating from b = to b = a, we find for the potential energy of the 

 complete disc of radius a, 



Thus, from equation (324), 



- B = 7v" : '- 



or, since cr = 



Pr ' 37V 

 rl 



Thus an upper limit to the whole resistance is 



lr_ _8r_ 

 Tra 2 . 37r 2 a' 



o 



which is the resistance of a length I +^ a of the tube. 



O7T 



Thus we may say that the resistance of the whole is that of a length 



Q 



I + aa of the tube, where a is intermediate between - and ^ , i.e. between 



4 O7T 



785 and "849. Lord Kayleigh*, by more elaborate analysis, has shewn that 

 the upper limit for a. must be less than "8242, and believes that the true 

 value of a must be pretty close to '82. 



THE PASSAGE OF ELECTRICITY THROUGH DIELECTRICS. 



395. Since even the best insulators are not wholly devoid of conducting 

 power, it is of importance to consider the flow of electricity in dielectrics. 



Using the previous notation, we shall denote the potential at any point 

 in the dielectric by V, the specific resistance by r, and the inductive capacity 

 by K. We shall consider steady flow first. 



If the flow is to be steady, the equation of continuity, namely 



SV\ 3/1SFN 3/13FN ( ^ }> 



must be satisfied. Also if there is a volume density of electrification p, the 

 potential must satisfy equation (62), namely 



ffeir)-!-* < 326 >- 



dz\ dzj 



From a comparison of equations (325) and (326), it is clear that steady 

 flow will not generally be consistent with having p = 0. Hence if currents 

 are started flowing through an uncharged dielectric, the dielectric will 

 acquire volume charges before the currents become steady. When the 



* Theory of Sound, Vol. n. Appendix A. 



