234 Mr. J. J. Thomson on the Electric and Magnetic 

 .'. T, the kinetic energy due to the electrification 



-#(»*+•*+■*)*** 



2fieY- 



~ 15a * 



Hence, if m be the mass of the sphere, the whole kinetic 

 energy 



-(;+i£)^ (3) 



or the effect of the electrification is the same as if the mass of 



4 ixe 

 the sphere were increased by =-=- — , or, if Y be the potential 

 a lo a 



of the sphere, by =-= >u<K 2 V 2 a. 

 JLO 



To form some idea of what the increase of mass could amount 

 to in the most favourable case, let us suppose the earth elec- 

 trified to the highest potential possible without discharge, and 

 calculate the consequent increase in mass. According to Dr. 

 Macfarlane's experiments, published in the Philosophical 

 Magazine for December 1880, the electric force in air at ordi- 

 nary temperatures and pressures must not exceed 3 x 10 12 

 (electromagnetic system of units). The electric force just 

 outside the sphere is V/a ; hence the greatest possible value 

 of V is 3 X 10 12 a, where a is the radius of the earth. Putting 



this value for Y, /a=1, K= q . in2 o ; a=6'4x 10 8 , we get for 



the corresponding value of the increase of mass 7 x 10 8 grms., 

 or about 650 tons, a mass which is quite insignificant when 

 compared with the mass of the earth. 



For spheres of different sizes, the greatest increase in mass 

 varies as the cube of the radius; hence the ratio of this increase 

 to the whole mass of the sphere is constant for all spheres of 

 the same material ; for spheres of different materials the ratio 

 varies inversely as the density of the material. 



If the body moves so that its velocities parallel to the axes 

 of x, y, z respectively are p, q, r, then it is evident that the 

 effect of the electrification will be equivalent to an increase of 

 4 

 — fMK 2 Y 2 a(p 2 + g 2 + r 3 ) in the mass of the sphere. 



§ 3. To find the magnetic force produced by the moving 

 sphere at any point in the field. By equations (2) we have, 

 for points outside the sphere, 



