206 CALCULATION OF INERTIA [APP. A. 



The strength of the magnetic field will be similarly 

 modified in this case ; but the simplest mode of stating that 

 is to express it in terms of E, and to say that always 



H = K^U sin 0. 



The rate of transmission of energy will be the vector 

 product of E and H ; and the whole magnetic energy, that 

 is the whole kinetic energy due to the current, i.e., due 

 to the motion will be obtained by integrating the ordinary 

 expression yuH 2 /87r, all over space outside the charged 

 sphere, viz., from a to oo all round. Its charge is assumed 

 to be superficial, so that no energy is inside. In the general 

 case this expression is a little long, but in the most im- 

 portant case, when the speed of motion u is decidedly less 

 than the speed of light v, it is quite simple, and the 

 working may as well be given: 



Kinetic energy 



f fsi 

 J J J_ 



COS 2 0-1 



Comparing this with mechanical kinetic energy, 

 we see that the charge on the sphere confers upon it 

 additional kinetic energy, as if its mass were increased on 

 account of the charge by the amount 



2/ze 

 m= 

 3a 



This may also be written 



2 



o~2 



3t> 2 



2 /UiK g 2 2 e 2 

 ra = ^ = o~2 e = ^~9 x charge x potential, 



2 2 



Q 



or, -r rai> 2 = the electrostatic energy of the charge : 



supposed a spherical shell of electricity. 





