17:? Induction of Current* in Continuous Media [on. xv 



Infinite /Yr/w Current-sheet. 



540. Lot the current-sheet bo of infinite extent, and occupy the whole 

 of the plane of .rj. and lot the moving magnetic system bo in the region 

 in which : is negative. Then throughout the region lor which t is positive 

 the potential 1} -i I!' has no poles, and henee tin 1 potential 



has no poles. Moreover this potential is a solution of Laplace's equation, 

 and vanishes over the boundary of the region, namely at infinity and o\ er 

 the plane : = (c\\ equation (-1-S7 N )V Hence it vanishes throughout the 

 whole region ^cf. ^ lSo\ so that i^jnation (4S7) must be true at every point. 

 in the region for whieh : is ]>ositive. We may accordingly integrate with 

 respeet to j and obtain the expiation in the form 



no arbitrary fnnetion of ./. // beini;- added beeause (lie etpiation must be 

 satisfied at infinity. 



The motion of the system of magnets on the negative side of the sheet 

 may be replaced, as in .">:>}). by the instantaneous creation of a number of 

 poles. At the creation of a single pole currents are set up in the sheet such 

 that i}-fi- remains unaltered (cf. equation (4S!^ on the positive side of 

 the sheet. Thus these currents form a magnetic screen and shield the space 

 on the positive side of the sheet from the effects of the magnetic changes on 

 the negative side. 



To examine the way in which these currents decay under the influence 

 of resistance and self-induction, we put <}' = in equation (4SD). and find 

 that 1} must be a solution of the equation 



</n a an 



</r ~:27ra7' 

 The general solution of this equation is 



and this corresponds to the initial value 



n =/(*, //. 



Thus the decay of the currents can be traced by taking the field of 

 potential H at time t = and moving it parallel to the axis of z with a 



velocity ~, 



REFERENCES. 



J. J. THOMSON. . \f t!,c Mathematical Theory of Electricity and Magnetism, 



xi. 



MAXWKI.L. Electricity a rum, Part iv. Chap. xil. 



