304 BELL SYSTEM TECHNICAL JOURNAL 



First, it should account for the field due to the magnet at every 

 point outside; 



Second, its value at every point inside the magnet should be a definite 

 function of the thing which we have just tentatively defined as "the 

 magnetic field" at that point; viz. the resultant of that field which 

 the external agencies would produce were the magnet away, and that 

 which the magnetization / should itself produce. 



Or, in other words: it should be possible to build up a reproduction 

 of the magnetized piece of iron out of little magnets, the magnetic 

 moment of each depending in a perfectly definite way on the force 

 exerted on it by the other little magnets and by the external world, 

 and all together producing the same effects in the external world as 

 the piece of iron does. 



In saying "it should be possible" I do not mean to imply that there 

 is an obligation resting upon Nature to construct magnetizable 

 objects in such a way that it is possible. One could not prove a 

 priori that she does. One must take variously shaped pieces of 

 magnetizable metal and observe their behavior in various impressed 

 fields, and ascertain for each whether or not there is a function /. 

 In so doing, one is liable to encounter very great mathematical diffi- 

 culties. In fact, the difficulties are likely to prove insuperable unless 

 the piece of metal is shaped in one or other of a few definite ways, 

 and the impressed field is uniform and properly oriented. 



Let us attack the problem from the other side, and enquire first 

 whether it is possible so to shape a piece of iron and so to orient 

 the impressed field, that the extra field due to the magnetization 

 should vanish everywhere within the iron, and the actual field should 

 everywhere be identical with the impressed field — so that although 

 there is a function / differing from zero, yet Hi = and H = He 

 everywhere inside the iron. This condition would be realized, if one 

 could make an infinitely long straight rod and expose it to an infinitely 

 extended uniform field parallel to its axis. It is very nearly realized 

 along the middle of a wire several hundred times as long as it is thick, 

 set parallel to the earth's field or along the axis of a solenoid somewhat 

 longer than the wire itself. It is very nearly realized within the 

 substance of a ring-shaped piece of metal pervaded everywhere by 

 an impressed field following the curvature of the ring; a field of 

 this character can be produced by wrapping a current-carrying wire 

 around the ring. 



In these cases, or rather in the ideal cases to which these are close 

 approximations, the vectors He and / are uniform throughout the 

 metal; the relation between their magnitudes is the relation between 



