Motion of a Perfectly Conducting Electrified Sphere. 641 



spherical polar coordinates is also adopted, following a very 

 kind suggestion from Dr. Bromwich. 



The general method consists in imparting to the sphere, in 

 a manner which will hereafter appear, a uniform acceleration 

 and deducing the initial field purely from geometrical con- 

 siderations. The effective force on the sphere is then calcu- 

 lated, and the coefficient of the acceleration in the expression 

 for this force is taken as representing the electromagnetic 

 mass of the sphere. 



I. When the sphere starts from rest. 



The sphere is perfectly conducting of radius a, with a 

 total charge e, and the acceleration is s, applied in a direction 

 which is taken as the polar axis of the coordinates, the centre 

 of the sphere coinciding initially with the origin. 



The acceleration is considered so small that the displace- 

 ment of the sphere in the time taken by radiation to travel 



so? 

 across the sphere is small compared with the radius ; —j is 



small compared with a. 



We have obviously only to deal with a case of symmetry 



about the polar axis. The Maxwell equations for the field 



outside the sphere can then be written 



c 2 dt K ' J \ r 2 "dfi' r sin O^r J 



dt-r\dA rl) ~d6 J 3 



(X, Y, Z ; a, j3, 7) being the usual components of the 

 electric^and magnetic vectors along the polar directions, and 

 /i= cos ^ and yjr = ry sin 6. This leads as usual to the 

 equation for i|r, 



1 tfyfr dV 1-/1-B 2 ! 



+ 





c 2 dt 2 - ?> 2 T r 2 ~dr 2 ' 

 of which the known general solution is 



♦—■_(- 'D'R^] <-<•■> 



The solution being restricted to the only necessary case of 

 expanding waves. 



We now attempt to find the field outside the sphere at the 

 end of a time t, to the first order in the acceleration. The 



small displacement of the centre of the sphere is f= *— . and 

 the equation to its surface is 



r = a + f cos 6. 



