FORCE IN THE INTERIOR OF A MAGNET. 307 



321. FORCE IN THE INTERIOR OF A MAGNET. We cannot 

 determine the magnetic action in the body of a magnet itself without 

 making a cavity in which a small test magnet may be placed ; but 

 the creation of a free surface in the interior of a magnet is equivalent 

 to the formation of a surface layer having in general a finite action 

 on 'the points which it contains, and this action depends on the 

 form of the cavity. 



A mass placed in the cavity is then outside the acting masses, 

 and the force which it undergoes may be determined in the usual 

 manner. Tfris force is the resultant of two others, one due to 

 external masses and the other to the surface layer of the cavity ; the 

 second force depends on the shape and orientation of the cavity 

 while the former is independent of it. 



The strength of magnetisation may, moreover, be regarded as 

 constant in magnitude and in direction, throughout the whole extent 

 of the infinitely small magnet which is removed ; it would therefore 

 be possible to determine the second force for certain simple forms 

 of the cavity. 



322. Let us consider, in the first place, a cylindrical cavity 

 the generating lines of which are parallel, and the bases at right 

 angles, to the strength of magnetisation. The density of the fictive 

 layer will be null on the lateral "walls, since the perpendicular com- 

 ponent of the magnetisation is zero at every point ; on the two bases 

 which are perpendicular to the magnetisation, the density will 

 be uniform, equal to + I on one and - I on the other. If the 

 extent of the base is equal to a, there will be equal and contrary 

 magnetic masses + al and - al at the two ends of the cylinder. 



Let us imagine the cylinder to be circular : let r be the radius 

 of the base and zh the height ; the action of two layers on a point 

 in the middle of the axis is double that of a homogeneous disc on a 

 point of the perpendicular raised at its centre. In order to calculate 

 this action, let us suppose the density equal to unity and the disc 

 divided into concentric elementary zones ; the component along the 

 axis of the action of one of these zones at a distance p from the 

 point in question is 



h 



or, taking into account the relation p 2 = r 2 + /^ 2 , 



#w*, 



X 2 



