ACTION OF A RIGID MAGNET AT EXTERNAL POINTS 61 Let us consider now, instead of a simple bar-magnet, a ferromagnetic body of any shape, magnetised in any arbitrary manner without reference to the cause of such magnetisation. When we wish to express this total independence of the magnetisation on external causes, we shall speak of such a body as a rigid magnet; the phenomenon of magnetic retentiveness shows us the possibility of realising such a rigidly magnetised body with some degree of approximation. Within the region occupied by the body, therefore, we are to consider the distribution of the vector 3 to be entirely arbitrary (and in general, consequently, complex-solenoidal, § 36); at the bounding surface the component along the normal drawn inwards is 3,, while along the normal drawn outwards into the magnetically indifferent surrounding medium, the component of magnetisation is zero. To calculate the magnetic intensity at external points, we may divide the body up into elementary parallelopipeds de dy dz, write down the expression for the effect due to the three pairs of opposite faces of X an infinitely small paral +Z FIG. 11 r+dr lelopiped, and integrate this expression throughout the whole volume of the body. Let us consider the parallelopiped dx dy dz as a short bar parallel to the axis of z; then the component of magnetisation in a direction parallel to this axis becomes +3. Since the magnetisation is a vector, we make what use we please of the principle of resolution into components. Let us now fix our attention on the upper face of the parallelopiped (shaded in fig. 11), its co-ordinates being x, y, z, and the area of the face in question de dy; its magnetic strength, which determines its influence at external points, is therefore + 3, dx dy, by the definition already given (§ 19). At a point P (, n, ), whose distance from (x, y, z) is r, this end-element will exert the magnetic intensity 5 given by the equation $ = + 3, dx dy Before combining the field due to the above end-element face with that due to the opposite face of the infinitesimal parallelopiped, let us further investigate the character of the distribution of the vector §. § 47. Distribution of Magnetic Intensity. We shall first prove that the magnetic intensity is everywhere lamellarly distributed, and to this end we must form the derivative with respect to of one of the components given above, for example the Y-component. We have Inserting this value in the above equation, we obtain The two derivatives are identical, and continue to be so, even when the value of r is reduced to any assignable extent. same way we may show that just as In the DISTRIBUTION OF MAGNETIC INTENSITY 63 The equations (7) of § 39 are consequently fulfilled, so that everywhere without exception throughout the whole field the distribution of 5 is lamellar; even at the place where the end-element itself is situated. In the next place, let us form the derivatives of the components of magnetic force with respect to the co-ordinates of the point P, by differentiating the expressions given in § 46 with respect to E, n, ; thus we find while for the sum of the right-hand terms of these equations we have the expression 3. dx dy 3 r2 — 3 (§ — x)2 — 3 (n − y)2 — 3 (5 — z)2 доб Referring to (18), this expression is seen to be zero for finite values of r, while for infinitely small values of r it assumes the form 0/0, and moreover it can be shown that in the latter case the expression does not vanish. Thus we have that is, the distribution of § is solenoidal throughout the field, except in the place where the end-element itself is situated. In accordance with the law of superposition VII (§ 43), we can now extend by summation to any number of elements the property just proved for a single end-element. Thus we arrive at the following theorem : X. The magnetic intensity due to a rigid magnet arbitrarily magnetised has everywhere a lamellar-solenoidal distribution, except in those places where end-elements are situated; there the distribution is only lamellar. § 48. Potential of a Rigid Magnet.-In accordance with what we have just seen, the magnetic intensity due to a rigid magnet has everywhere a scalar potential, which satisfies Laplace's equation at every point, except in those places mentioned in Theorem X. This function is called the magnetic potential of the rigid magnet, and, as before (§ 45), we shall denote it by T. Let us now turn our attention once more to the magnetic field due to the elementary parallelopiped. Let the face which has hitherto been considered (that which is shaded in fig. 11, p. 61) be distinguished by the number 1, the face immediately opposed to it by 4; and in the same way let the two remaining pairs of opposite faces be numbered 2 and 5, 3 and 6, respectively. We shall denote by 8 T, the magnetic potential at the point P, due to the face 1, and from the general theory of potential it immediately follows that Consider next the elementary face 4; let its distance from P ber+dr; its strength is 3, de dy. It produces at P a magnetic potential & T1, which is found exactly as in the case of 1; we have 4 If we add together the two terms &T, and ST, of the potential, we obtain the magnetic potential due to the pair of faces (1.4), which we denote by 8 T1.4, and which has accordingly the following value: On comparing this expression with that for 8T, or & T1, it will be seen that the magnetic potential of a pair of opposite end-elements is an infinitesimal of the third order, while that POTENTIAL OF A RIGID MAGNET 65 of a single face belongs only to the second order of small quantities. When we pass from the face 1 to the face 4 (compare fig. 11, p. 61), the Z-co-ordinate increases by -dz, and consequently Substituting this value in the expression for &T1. we obtain 1.4 The pairs of opposite faces, 2 and 5 (strength ± 3, dy dz) and 3 and 6 (strength 3, dz da) are to be dealt with in exactly the same way. We obtain, then, by summation as the value of the magnetic potential at P, due to the whole parallelopiped, ST = ST1.4 + ST2.5 + ST3.6 If in this equation we insert the expression (20) and the corresponding expressions for 8T2.5 and 8T3.6, the result obtained To find the potential at P due to the entire rigid magnet, &T must be integrated over the entire volume of the latter. If we now apply the principle of integration by parts to each term of the above integral separately, putting in the first term, |