68 ELECT EICI T Y [ELECTEOMAONETISM. Linear circuit in magnetic field. shell. In passing from P to Q, without cutting the shell, the solid angle w decreases by ITT infinitely nearly. N"ov, (faring the passage from Q to P we may not represent ths actiju of the current by S, but nothing hinders us from representing its actiou by another shell S , which does not pass between Q and P, but is at a finite distance from either of them ; for it will be remembered that the shell which represents the action of a current i is definite to this extent merely that its strength is i, its boundary is the circuit, and it does not pass through the point at which the action is being considered. But infinitely little work, owing to the action of S , is done in passing from Q to P. Hence the work done by a unit pole in going once completely round any path which embraces the current once is 47rt. To reconcile this result with the continuity of the mag netic potential of a linear circuit, for the existence of which we have now furnished sufficient evidence, we must admit that the potential of a linear circuit at any point P is V = /(w + 47r), where n is any integer. In other words, V is a many-valued function differing from i times the solid angle subtended at P by a multiple of ^TTI. If we pass along any path from P and return thereto, the difference of the values of V, or the whole work done on the journey, is zero if the path does not embrace the circuit, imri if it embraces 1 it n times The considerations enable us to determine the action of any closed current on a magnetic pole, and consequently on any magnetic system. We have next to find the action on a linear circuit when plaed in any given magnetic field, whether due to magnets or electric currents. This we do by replacing the circuit acted on by its equivalent magnetic shell. If the potential at any point of the magnetic field be V, then the potential energy of a magnetic shell S, of strength i, placed in the field is given by JJ dx dy i where (I, m, n) are the direction cosines of the positive direction (south to north) of the normal to the element dS. Since, so long as the magnetic force considered is not due to S itself, there is none of the magnetism to which V is due on S, we may write - a, b, dV dV dV - c for -j- > -y- > -j- where a, b, c are the components of the magnetic induction. 8 Then, if N = ff(la + mb + nc)dS (i.e.,=the surface integral of magnetic induction, or the number of lines of magnetic force which pass through the circuit), we may write M=-zX (4). From this expression for the potential energy of the equivalent magnetic shell we can derive at once the force tending to produce any displacement of the circuit regarded as rigid. Thus let $ be one of the variables which determine the position of the system, then the force * tending to produce a displacement </$ is given by *rf$> + dM = 0, or .dN
- ~;r7- (5)-
d<j) Hence the work done during any displacement of a closed circuit, in which the current strength is i, is equal to i times the increase produced by the displacement in the number of lines of force passing through the circuit. The force tends, therefore, to produce the displacement or to resist it, according as the displacement tends to increase or to diminish the number of lines of force passing through the circuit. It is evident, therefore, that a position of stable equilibrium will be that in which the number of lines of magnetic force passing through the circuit is a 1 On the space relations involved here see Maxwell, vol. i. 17, &c. 8 Magnetic induction is used here in Maxwell s sense. It coincides in meaning with "magnetic force" at points where there is no mag netism. " Line of force " in Faraday s extended sense is synonymous with " line of induction" in Maxwell s sense. maximum. If that number is a minimum, we have a case of unstable equilibrium. Maxwell 3 has shown how we may deduce from the Action above theory the force exerted on any portion of the circuit on e!e - which is flexible or otherwise capable of motion. "If a n . lc ^ t ; 0; portion of the circuit be flexible so that it may be displaced independently of the rest, we may make the edge of the shell capable of the same kind of displacement by cutting up the surface of the shell into a sufficient number of portions connected by flexible joints. Hence we conclude that, if by displacement of any portion of the circuit in a given direction the number of lines of induction which pass through the circuit can be increased, this displace ment will be aided by the electromagnetic force acting on the circuit." From these considerations we may find the electromagnetic force acting on any element ds of the circuit. Let PQ (fig. 30) ba the element ds belong ing to the arc AB of any circuit. Let P$ be the direction of the magnetic induction 4 at P, and $ its magnitude. It is obvious that no motion of PQ in the plane of PQ and P$ will increase or diminish the number of lines of force passing through the circuit ; con sequently no work will be done in any such dis placement. Hence the resultant electromagnetic force R must be perpen dicular to the plane QP$. Let PR be a small displacement perpendicular to this plane, the work done in the displacement is R.PR, and the number of lines of force cut through is i times the rectangular area PQR multiplied by the com ponent $ sin 6 of the magnetic induction perpendicular to it. Hence we have RxPK-iefoxPRx Hence the resultant electromagnetic force on the ele ment ds may be determined as follows : Take P^ in the direction of the resultant magnetic induction (magnetic force) and proportional to i$, and take PQ in the direction of ds and proportional to it; the electromagnetic force 5 o:: the element of the circuit is proportional to the area of the parallelogram whose adjacent sides are P|3 and PQ, and is perpendicular to it. The force in any direction making an angle < with the direction of the resultant is of course Rcos</>. The following consideration is convenient for determining which way the resultant force acts. It is obvious that the force on the element will be the same to whatever circuit we suppose it to belong, so long as the direction and strength of the current in it is the same. Take, then, a small circuit PQK perpendicular to the line.-i of magnetic induction (magnetic force) near PQ, in such a way that the direction of the current in PQK (as deter mined by the direction in PQ) is related to the direction of the magnetic induction in the same way as rotation and translation in right-handed screw motion ; then the ele ment PQ tends to move so that the number of lines o f force passing through PQK increases. 3 Electricity and Magnetism, vol. ii. 490. 4 " Resultant magnetic force," if there is none of the magnetism pro ducing it at P. We need scarcely remind the reader that this is a ponderomotiv: force acting on the matter of the element of the circuit. There is r.j question of force acting on the current or the electricity in it. c From this may be derived the following, which is often very con venient. Stand with feet on PQ and body along the positive direction of the line of magnetic force and look in the direction of the cur-
j rent, then the forte is towards the right hand.