MAGNETISM 229 ?ase of
- olenoid-
,1 mag- ietiza ; ion. at every point external to the body is therefore equal and opposite to that of the internal electricity. If, therefore, we change the sign of the surface density at every point, we obtain a surface distribution whose potential at every external point is the same as that of the body. There is of course only one such distribution : we may call it Gauss s distribution. Poisson s distribution will coincide with that of Gauss provided the magnetization be such that lesnlt- .nt force nside a aagnet- zed iodv. lag- letic in- luction. when this condition is satisfied at every point of the body, A is said to be " solenoidally " magnetized; a particular case is that of uniform magnetization. So long as the point considered is external to the magnet there is no difficulty in attaching a definite meaning to the resultant magnetic force (|p) at a point; its components are given by dx dy dz and the values obtained will be the same whether V be calculated by means of Poisson s or of Gauss s distribution. Inside the body the result is otherwise, for reasons that are not difficult to understand, when we examine the nature of our fundamental assumptions. It is therefore necessary to be careful to define what we mean by resultant magnetic force in the interior of a magnet. It is denned by the above equation (9) on the understanding that V is calcu lated from Poisson s distribution. We can show that |p thus defined is the resultant force in an infinitely small cylindrical cavity within the magnet, whose axis is parallel to the line of magnetization, and whose radius a is infinitely small compared with its axis 25. The removal of the matter filling such a cavity will affect Foisson s volume distribution to an infinitely small extent ; the alteration of the force if any, will therefore arise simply from the surface distribution which we must place on the walls of the cavity in order to make up the complete representation of the action of the magnet in the cavity. This distribution reduces to two circular disks of radius a at the two ends, the densities of the magnetism on which are - I and + 1 respectively. The action due to these is a force 47rl(l - &/V 2 + b 2 ) in the direction of magnetization. If a be infinitely small compared with b, this force becomes zero, which proves our proposition. If, on the other hand, the cavity in the magnet be disk- shaped say a narrow crevasse perpendicular to the line of magnetization then the force due to the distribution on its walls becomes 4-rrI, and the resultant force in the cavity is no longer <p, but a force |8, whose components are ^ is called the " magnetic induction " at the point (x, y, z). From the definition of |j it follows that outside the magnet . (11). Inside da d/3 dy r ^ I i n 7 1 ; r , U dx dij dz da, d/3 dx dy dy U. dE dx dy - dz (12). Con- tinuity and discon tinuity of $ and At the surface of a magnetized body the tangential com ponent of |fj is continuous, but the normal component increases abruptly by 47rlcos# in passing from the inside to the outside of the surface. Outside magnetized matter the magnetic force and the magnetic induction are coincident. Inside we have da, db dc I [ . dx dy dz dy dz (13). dy dz " dx Hence the magnetic induction satisfies the solenoidal con dition both inside and outside magnetized matter. It has normal continuity, and, in general, tangential discontinuity, at the surface of a magnetized body. For if v, r and n, t be the normal and tangential components of $) and |3 just inside, and v , T and n , t the corresponding components just outside the surface near any point, we have n = v + 4-n-l cos 6, and n =v ; but v = v + iirleos6, therefore n = n . On the other hand r =t f , whereas T is the resultant of t and 47rlsin0, which is parallel to the surface, but otherwise may have any direction according to circumstances ; hence, since t = t, in general T is not equal to r. In fact there will be tangential discontinuity of the magnetic induction unless the line of magnetization be perpendicular to the surface of the magnet ; in this case there is complete continuity of the magnetic induction. When the magnetization at the surface is tangential, there is, on the other hand, complete continuity of the magnetic force. It follows from the above that the surface integral of Surface the magnetic induction taken over any closed surface S integral vanishes. of mag- First, let the surface be wholly within or wholly without con- *j e tinuously magnetized matter. We have, integrating all over S and ^ all over the space enclosed by S, the analytical theorem db dc du dz (14); hence the result follows, for every element of the right-hand integral vanishes. Next, suppose S to be partly within and partly without a magnetized body. Divide it into two parts by a double partition one of whose walls runs outside the surface of the body and infinitely near it, the other inside and infinitely near it ; then, on account of the normal continuity of $, the surface integral will be the same in absolute value over each of these walls. Hence the integral over the whole of S differs infinitely little from the sum of the integrals over the two surfaces into which it is broken up by the double partition, each of which vanishes by the former case. Hence the theorem holds in this case also. We may therefore apply to lines and tubes of magnetic induction without restriction all the theorems proved for lines and tubes of electric force in space free from electrified bodies. We may speak of the number of lines of magnetic induction instead of the surface integral if we choose. And we have this important theorem : The number of lines of magnetic induction that pass through an unclosed surface depends merely on its boundary. There must therefore be a vector ^, whose line integral Vector round the boundary is equal to the surface integral of 38 P teH - over the surface. The components F, G, H of i are connected with those of 8 by the equations tial. dK dy dtt dz _ dz dx clG dx as has been shown in the article ELECTRICITY, vol. viii, p. 69. Mutual potential energy and mutual action of tiuo magnetic, Mutual systems. The potential energy of a small magnet is fcCVj-"^), potential where Vj and V 2 are the values of V at its negative and positive poles, energy If the magnet be infinitely small, of length ds say, the direction of two cosines of ds being A,fj.,v, this may be written icdsdV/ds, i.e., mdV/ds, magnet?. or, if we are considering a magnetized element of volume dv,
h M-V- + "-T- }dv
^ dy dz J dx dV dV - i c (16). Hence the potential energy of the whole magnetic system in a field whose potential is given by V is W = +E^ + C~}dv dx dy dz -M the integration being extended all over the magnetized masses sup posed to be acted upon. Integrating by parts, we get at once (18), ff and p being the surface and volume densities of Poisson s distri bution, a result that might have been expected. W may also bo expressed as a sextuple integral ; for, if I , A , yu , j , x , y 1 , z 1 refer to the acting system, then > d > ju -r-> + "
^ dy