425-428] Force inside a Magnetised Body 371 



427. One way of measuring the force at a point Q inside a magnet will 

 be to imagine a cavity scooped out of the magnetic matter so as to enclose 

 the point Q, and then to imagine the force measured on a pole of unit 

 strength placed at Q. This method of measurement will only determine a 

 definite force at Q if it can be shewn that the force is independent of the 

 position, shape and size of the cavity, and this, as will be obvious from what 

 follows, is not generally the case. 



428. Let us suppose that, in order to form a cavity in which to place 

 the imaginary unit pole, we remove a small cylinder of magnetic matter, the 

 axis of this cylinder being in the direction of magnetisation at the point. 

 Let this cylinder be of length I and cross-section S, and let the intensity of 

 magnetisation at the point be /. Let the size of the cylinder be supposed to 

 be very great in comparison with the scale of molecular structure, although 

 very small in comparison with the scale of variation in the magnetisation 

 of the body. 



In steel or iron there are roughly 10 23 molecules to the cubic centimetre, so that a 

 length of 1 millimetre may be regarded as large when measured by the molecular scale, 

 although in most magnets the magnetisation may be treated as constant within a length 

 of a millimetre. 



At a point near the centre of this cavity we are at a distance from the 

 nearest magnetic particles, which is, by hypothesis, great compared with 

 molecular dimensions. Hence, by 416, we may regard the potential at 

 points near the centre of the cavity as being that due to the following 

 distributions of imaginary magnetic matter: 



I. A distribution of surface- density lA + mB + nC, spread over the 

 surface of every magnet. 



II. A distribution of volume-density 



_ dB dC 



spread throughout the whole space which is occupied by magnetic matter 

 after the cavity has been scooped out. 



III. A distribution of surface-density lA+mB + nC, spread over the 

 walls of the cavity. 



From the way in which the cavity has been chosen, it follows that 

 I A + mB + nC vanishes over the side-walls, and is equal to + / on the 

 two ends. 



The force acting on an imaginary unit pole placed at or near the 

 centre of the cavity may be regarded as the force arising from these 

 three distributions. 



242 



