2.30 Whence / , d X A/-J-; V cte , + U - , d 1 + > -j-; l-=r- dz J D (19). A remarkable expression for W may be obtained by supposing the integration in (17) extended throughout the whole of space, on the understanding that A, B, C are zero where there is no magnetized matter, and then integrating by parts. We get, since the surface integral at infinity may be shown to vanish, W CO -fff^ ^A dV, dC J-+-T--I--J- dx ay dz (20), where it must be understood that A, B, C vary continuously, how ever rapidly. In point of fact, where, as at the surface of a magnetized body, there is discontinuity, a finite portion of the integral will arise from an infinitely thin stratum near the surface. The proper representation of this part will be a surface integral, as may be seen by referring to (18), from which we might have started. If now V be the potential of the magnet acted upon, then whence d-V d?V d 2 V 7 n ~T 7 .1 1 7 >> dx- ay- dz- 1 dA. dE dC -. I ^ H 7 dx ay dz dV dV o!V dT_ dV dV dx dx dy dy dz dz RR cos0 dv (21), where R and R are the resultant forces at any point of space due to the acting and acted-upon systems respectively, and the angle between their directions. Force In practice W is expressed as a function of the variables (equal in and number to the degrees of freedom) that determine the relative couple position of the two systems ; differentiation with respect to any one on a of these then gives the generalized force component tending to magnet decrease that variable. in given We may also calculate the forces directly. For, the components field. of force on the element dv, being the differences of the forces acting on the two poles of the element, arc f^+BS + oSW*^
dx dy dz J
ami the components of couple, in calculating which the field may be supposed uniform, are (see above, p. 228) (76 - fitydv, &c. Hence, integrating, we get, with the chosen origin, for the com ponents of the whole force and couple, dz + B 4-C I > (22) tj f]i. ~7~ / * and similarly for || and dy dz dx dy i and similarly for ,3ft and |t. In the important case of a uniform field whose components are a, /8, y, we have W = - K(/a + mff + ny) (23), K being the moment of the magnet, and I, m, wthe direction cosines of its axis. From this formula the results given above (p. 227) can be deduced with great ease. Examples. Some examples of the application of the foregoing theory are here given, partly on account of their intrinsic value as types enabling us to conceive the differ ent varieties of magnetic action, partly for the sake of the light they throw on the theory itself. The reader who i See Thomson, Reprint of Papers on Electricity and. Mactnetism, p. 433. magneti zation. desires more such should consult Maxwell s Electricity and Magnetism, or Mascart and Joubert, Lemons sur I Electricite et le Magnetisme. Solenoidal Magnets have already been defined as such that the Sole- vector I satisfies the solenoidal condition noidal dA. rfB dC i t Q dx dy dz The lines of magnetization, therefore, have all the properties of lines of magnetic induction or electric force. In particular, if we consider a portion of the magnet enclosed by a tube of the lines of magnetization, the product of the intensity of magnetization by the section at each point is the same. Such a portion of magnetized matter taken by itself is called a "magnetic solenoid," and the pro duct mentioned is called its "strength." It is clear (from thegeneral definition, or it may be proved directly from the secondary property just mentioned) that the action of the solenoid may be represented by the distribution of a certain quantity col of positive magnetism on the one end and an equal quantity of negative magnetism on the other, I being the intensity of magnetization, o> the normal sec tion at the end. The action therefore depends merely on the strength of the solenoid and on the position of its ends. The shape of the intervening portion is immaterial. If we suppose it straight, Equipo- aud if the section be infinitely small so that the magnetism at the teutial ends may be regarded as condensed at two points, we have an ideal lines ani magnet of finite length. The equipotential lines of such a magnet lines of in any plane through its axis are of course given by the equation force. 1 1 r const. r r (24), where r and r are the distances of any point Pon the line from the poles. The equation to the lines offeree is easily obtained ; 2 for, if NP Ideal and SP (fig. 21) make angles and 6 with the axis of the magnet, bar and <j) and <p with the line of force, we must have magnet. sin fyjr- sin <f> /r 2 = ; hence, since sin = rdO/ds , sin <f> = r d6 /ds , we get de/r - de /r =*= ; i. e. , sin 6 d9 - sin O dO = ; which gives for the equation to a line of force cos 6 - cos 6 = const (25). We may imagine a magnet of this kind so long that the action Two of one of its poles may be altogether neglected at points which are like at a finite distance from the other. We thus effectively realize what poles, never occurs in nature, viz., a magnet with one pole only. If we place the like poles of two such magnets near^ach other, we get a field the equipotential lines and lines of force in any axial plane of which are given by the equations
r cosfl-t- cos tf = const. (27). The lines of force given by equations (25) and (27) may be traced in a diagram by means of the following simple and elegant construc tion 3 Draw two circles A and B, having equal radii and N and S respectively for centres ; produce the line NS both ways, and, starting from the centre, divide it into any number of equal parts ; through these draw perpendiculars to meet the circles A and B ; 2 The first mathematical investigations of the equation to the lines of force of an ideal magnet appear to have been made by Playfidr at the request of Robison, and by loslie, Geom. Analysis, 1821. They had previously been very carefully eonM lered from an experimental point of view by Lambert, Mem. de I Acad. de Berlin, 1766.
3 Koget, Jour. Ro-_: lust., 1S31.