NIPHER EXPRESSION OF ELECTRICAL RESISTANCE, ETC. 535 



On the Expression of Electrical Resistance in Terms 

 of a Velocity. 



By Francis E. Nipher.* 



If a spherical shell of radius r be charged with Q units of elec- 

 tricity, the density of electrification being p , the force dF over 

 any element ds of its surface will be 3 - p^ds. This force is di- 

 rected radially outward, and is due to the action of the electrifi- 

 cation Q on the quantity pds upon the element. 



If the radius r be diminished to r' , the energy of the electri- 

 fication will increase if Q remains constant, this increase in 

 energy being due to work done on the sphere by some external 

 source, causing the sphere to collapse. If the element ds sweeps 

 through a distance dr, the stored energy will be 



dE = dFdr . . . (i) 



in which both dF and dr are essentially negative. 



Substituting in (i) the above value of dF and remembering 



that P = — o 



and ds = r^ du ^ 



where dio is the solid angle subtended by the element ds^ we have 



jj^ Q,^ dr ■, 



dE = ^ ~^dio 



8 rr r^ 



where one integration is carried over the surface of the sphere, 

 and the other is carried inwards between the limits r and r' . 

 Performing the integrations, we have 



^'-^ = f (^-^] ■ - w 



But — is the energy of a sphere' of radius ^', charged with 



Q units of electricity, and hence the potential of the sphere on 

 itself between the limits r and r' is equal to the difference in its 

 initial and final energy. 



If the sphere were connected with the ground by a wire of re- 

 sistance (/?), the radius (r) might be changed in such a manner 



* Read March i;th, 1884. 



