Motion of an Electrified Sphere. 



57 



with the value o£ F 12 at any instant, as we shall eliminate P 12 

 from the equations. 



Let TJj, Ti, and U 2 , T 2 be the electric and magnetic 

 energies of the sphere when moving with the velocities u ± 

 and u 2 , and let Wi be the energy radiated when the velocity 

 u t is suddenly destroyed, and W 2 the energy radiated when 

 u 2 is destroyed. Since the force which stops the sphere does 

 no work, the total energy in the electromagnetic field is 

 unchanged. Before the sphere is stopped the energy is 

 Uv+Ti, and after it is stopped the energy is U + Wi, where 

 U is the electrostatic energy. Hence 



and 



Ui+T^Uo+.Wi 



U 2 + T 2 =U + W 2 . 



The values of W 1 and W 2 , which may be deduced from the 

 results of § 4 ; are given by 



-U.(¥+^ + ^ + ...). (32) 



W 2 =U (!log^- 2 ). . . (33) 



Since w 1 = » 1 r, we can write 



W 1= U,{ilog^-2}. . . (34) 



If we prefer to do so, we may express "Wi in terms of 

 im f 2 , by the formula U = -| . hn v 2 , where m is the electro- 

 magnetic mass for infinitesimal speeds. 



Table I.- 



-Values of — ^ 



U A 



n l* 



w x /u . 



» x . 



Wj/Uo. 



o-i 



000671 



06 



0-31049 



0-2 



002733 



0-7 



0-47800 



03 



006346 



0-8 



0-74653 



04 



011824 



0-85 



095565 



0-5 



019722 



09 



1-27160 



