430 CURRENTS AND MAGNETIC SHELLS. 



magnetic mass is displaced in the vicinity of a current, the equivalent 

 shell is being constantly deformed, and recedes before it without 

 ever being met. 



The analogy of the two systems becomes closer if instead of 

 considering the magnetic force of a shell we consider the induction. 

 We know, in fact, that magnetic induction is a continuous function 

 of co-ordinates, that the flow of induction is maintained throughout 

 the entire extent of an orthogonal tube, and that the force and 

 magnetic induction have the same value for any point outside the 

 magnetised media. As a particular case, the magnetic induction in 

 the thickness of a shell is identical with the force which would be 

 produced there, if the shell, while still retaining the same contour, 

 and the same magnetic power, were deformed in such a manner as 

 no longer to include the point in question, and this force is equal to 

 that of an equivalent current which went along the contour. This 

 is evidently the same also for any system of currents ; from which is 

 deduced the general law : 



Any system of dosed currents is equivalent to a magnetic system^ 

 and the action of currents at a point is identical with the induction^ at 

 the same point , of the equivalent magnetic system. 



452. RELATIVE ENERGY OF A MAGNETIC SYSTEM AND A 

 CURRENT. The potential of a current at a point P, is within a 

 constant equal to -Iw, if we denote by o> the solid angle under 

 which the negative face of the current is seen. The product - mlu 

 is the work which would be expended in bringing a magnetic mass 

 equal to m, from an infinite distance to this point, without traversing 

 a continuous surface bounded by the current. The potential energy 

 of the mass m at the point P is then, within a constant, equal 

 to mliD. 



If this mass has passed the surface of the current n times ,by the 

 positive face to arrive at the point P, the work ml^ir must each time 

 have been expended; the total work is then 



ml (^irn a>). 



Conversely, if the mass is left to itself, it tends to turn in- 

 definitely around the current, and at each turn expends an amount 

 of energy equal to m^I. 



This continuity of motion is not possible with two magnetic 

 systems, for the potential is then a determinate function of the 

 co-ordinates ; it would, moreover, be inconsistent with the principle 

 of the conservation of energy. With currents the movement may 



