166 Prof. J. J. Thomson on the 



field of magnetic force, where the components of the external 

 magnetic force are a , (3 , y , the momentum per unit volume 

 parallel to w is 



the rate at which this momentum enters a sphere concentric 

 with the moving sphere is 



S 



- {gyo—h(fi + ^7rw/)}zwd^, 



where d$ is an element of the surface of the sphere whose 

 radius is r. Integrating this we get 



V VVVio 2 w* ° (V—w)/ 



this is the increase in momentum per unit time, and therefore 

 measures the force on the space inside the sphere. It will be 

 noticed that this vanishes when V =-w : thus a charged sphere, 

 moving with the velocity of light through a magnetic field, 

 will not be acted upon by any force. We may regard the 

 force on the sphere as arising in the following way. Let us 

 suppose that we have a uniform magnetic field parallel to ?/, 

 then, in consequence of this external magnetic force, the 

 closed tubes of electrostatic induction will be moving about 

 in the field, the positive tubes going one way and the nega- 

 tive ones the opposite. Let us consider what happens when 

 one of these tubes go?s through the' sphere. If /3 X and /3 3 are 

 the values of the magnetic force where it respectively enters 

 and leaves the sphere, the momentum of the tube parallel 

 to x when it enters the sphere is proportional to /3 lf and 

 when it leaves the sphere to /3 2 . Now fii and (B 2 are different 

 because on one side of the sphere the magnetic force due to 

 the motion of the sphere acts in the same direction as the 

 external magnetic force, but on the other side of the sphere 

 it acts in the opposite direction. Thus the momentum of the 

 tube parallel to x is not the same when it leaves the sphere 

 as before it entered, so that the space inside the sphere must 

 have gained or lost momentum. 



The second case we shall consider is. that of the magnetic 

 field around a circular current of large radius. Then near a 

 portion of this circuit, which we may consider straight, the 

 tubes of electrostatic induction will be parallel to the circuit 

 and moving radially, the positive tubes {i.e. those parallel to 

 the direction of the current) moving inwards, the negative 

 ones outwards : the positive tubes move in to the conductor 

 and then contract in the manner, previously described. The 



