Therefore yoho+yn (where y is a scalar), and is thus perpendicular to the plane passing through the conductor and the extremity of the canal, and varies inversely as the distance of the latter from the conductor. This is exactly the observed effect of an indefinite straight current on a magnetic pole, or particle of free magnetism. 410. Suppose the conductor to be circular, and the pole nearly in its axis. Let EPD be the conductor, AB its axis, and C the pole; BC perpendicular to AB, and small in comparison with AE=h the radius of the circle. Let AB be a, i, AP = h (jx+ky) E BC= bk, B ∞ h fo {(h—by)i+a, xj +a,yk}, αλ (a2 + b2 + h2 — 2 bhy)ž -- where the integral extends to the whole circuit. 411. Suppose in particular C to be one pole of a small magnet or solenoid CC whose length is 27, and whose middle point is at G and distant a from the centre of the conductor. CGB A. Then evidently Let = a1 = a+lcos A, b = l sin A. Also the effect on C becomes, if a;+b2+h= A2, If we suppose the centre of the magnet fixed, the vector axis of the couple produced by the action of the current on Cis If A, &c. be now developed in powers of l, this at once becomes Thal sin A j{2 6 al cos A 15a2l cos2 A 312 + a2+h2 (a2 + h2)2 37'sin'A 15 h2 l2 sin'A + a2+h2 2 (a2+h2)2 Putting for that for pole C, couple 4 Thal sin A and changing the sign of the whole to get we have for the vector axis of the complete which is almost exactly proportional to sin A if 2a = h and be small. On this depends a modification of the tangent galvanometer. (Bravais-Ann. de Chimie, xxxviii. 309.) 412. As before, the effect of an indefinite solenoid on a, is Now suppose a, to be an element of a small plane circuit, & the vector of the centre of inertia of its area, the pole of the solenoid being origin. where A, and e, are, for the new circuit, what A and were for the former. Let the new circuit also belong to an indefinite solenoid, and let d. be the vector joining the poles of the two solenoids. 0 Two which is exactly the mutual effect of two magnetic poles. finite solenoids, therefore, act on each other exactly as two magnets, and the pole of an indefinite solenoid acts as a particle of free magnetism. 413. The mutual attraction of two indefinitely small plane closed circuits, whose normals are e and e,, may evidently be mutual action of the poles of two indefinite solenoids, making dò in one differentiation || € and in the other || €. But it may also be calculated directly by a process which will give us in addition the couple impressed on one of the circuits by the other, supposing for simplicity the first to be circular. Let A and B be the centres of inertia of the areas of A and B, and €1 vectors normal to their planes, σ any vector radius of B, AB = 3. for generally Von Seo = (Vno Seo + V.nV.0/Voo). Hence, after a reduction or two, we find that the whole force exerted by A on the centre of inertia of the area of B This, as already observed, may be at once found by twice These expressions for the whole force of one small magnet on the centre of inertia of another, and the couple about the latter, seem to be the simplest that can be given. It is easy to deduce from them the ordinary forms. For instance, the whole resultant couple on the second magnet may easily be shown to coincide with that given by Ellis (Camb. Math. Journal, iv. 95), though it seems to lose in simplicity and capability of interpretation by such modifications. 414. The above formulae show that the whole force exerted by one small magnet M, on the centre of inertia of another m, consists of four terms which are, in order, 1st. In the line joining the magnets, and proportional to the cosine of their mutual inclination. 2nd. In the same line, and proportional to five times the product of the cosines of their respective inclinations to this line. m 3rd. and 4th. Parallel to {M} and proportional to the cosine All these forces are, in addition, inversely as the fourth power of the distance between the magnets. |