SHUDDEMAGEN. — DEMAGNETIZING FACTORS FOR IRON RODS. 199 



the iron rod along the curved surface, as is well known. Now from the 

 mathematical theory we know that in the case of " soft " iron B, or ^x.H, 

 is a solenoidal vector, continuous throughout all space, whether iron 

 or air, not containing any fixed magnetic charges. Wherever lines of 

 induction leave the surface of the iron we must therefore have positive o- ; 

 for the vectors H and /, although not solenoidal in the iron, have always 

 the same distribution as the vector B, I is zero outside the iron, and 

 (T ^= I- COS (ji, I). This means that a part of the surface distribution 

 o- of the magnetism is closer to the middle of the rod than it would be 

 if / were uniform. There is also some magnetic matter in the form of 

 volume distribution p. This, however, does not materially influence the 

 argument, although it complicates matters somewhat. We shall come 

 back to the volume charge later. Therefore, as far as the surface mag- 

 netism is concerned, the demagnetizing force ^Hp is for every point P 

 actually greater than it would be if / were uniform. We thus reach 

 the result that the iV-curve has the end-points and K, but lies every- 

 where else to the right of the straight line OK. Indeed for the most 

 part the iNT-curve will be very decidedly to the right, for a very large 

 number of the lines of induction will leave the iron rod before reaching 

 the ends of the rod. The demagnetizing factor N^ is the minimum 

 value of N, although ^H^ is by no means vanishingly small. Near the 

 origin the ratio of H to I is comparatively large, although of course 

 still a fraction, so that according to (2) the / is more nearly uniform 

 than for higher points on the curve, so long as we do not pass the 

 point of maximum susceptibility, which is the point of tangency of a 

 line drawn from the origin to the normal magnetization curve ; therefore 

 the xY-curve is more nearly tangent to the line OK&t the origin than for 

 points a little more removed. As we increase H^ from to some point 

 Q whose /is of the order of /at Qq, the lines of magnetization increase 

 continually, but a larger and larger fraction of lines leave the rod be- 

 fore reaching the ends, and JV increases continually. Again, as we 

 follow the magnetization curve from any very large but finite value of 

 H' down toward Q, the /-lines spread out in greater and greater pro- 

 portion, and the AT increases for quite a long interval. This shows 

 that the curvature of the A''-curve changes sign at some point Q^, which 

 is a point of inflection for the A'-curve, and probably the only one- 

 We should expect, therefore, that the curve drawn in the second part of 

 Figure 3 on the / vs. NI plane represents roughly the qualitative be- 

 havior of an A"-curve for a finite rod. 



It remains to be shown that the volume distribution does not invali- 

 date the argument just given. From the theory of magnetism we know 

 that this can be expressed in the form 



