594.] THEORY OF A SLIDING PIECE. 217 594.] We have next to deduce from dynamical principles the expressions for the electromagnetic force acting on a conductor carrying an electric current through the magnetic field, and for the electromotive force acting on the electricity within a body moving in the magnetic field. The mathematical method which we shall adopt may be compared with the experimental method used by Faraday in exploring the field by means of a wire, and with what we have already done at Art. 490, by a method founded on experiments. What we have now to do is to determine the effect on the value of p, the electrokinetic momentum of the secondary circuit, due to given alterations of the form of that circuit. Let AA', BB be two parallel straight conductors connected by the conducting arc C, which may be of any form, and by a straight M A Fig. 38. conductor AB, which is capable of sliding parallel to itself along the conducting rails AA' and BB'. Let the circuit thus formed be considered as the secondary circuit, and let the direction ABC be assumed as the positive direction round it. Let the sliding piece move parallel to itself from the position AB to the position A'B'. We have to determine the variation of p, the electrokinetic momentum of the circuit, due to this displacement of the sliding piece. The secondary circuit is changed from ABC to A'B'C, hence, by Art. 587, p(ABC)-p(ABC) = p(AA′B′B). (13) We have therefore to determine the value of p for the parallelogram AABB. If this parallelogram is so small that we may neglect the variations of the direction and magnitude of the magnetic induction at different points of its plane, the value of p is, by Art. 591, B cos n. AA'BB, where B is the magnetic induction, * Exp. Res, 3082, 3087, 3113. η and the angle which it makes with the positive direction of the normal to the parallelogram AA'B'В. We may represent the result geometrically by the volume of the parallelepiped, whose base is the parallelogram AABB, and one of whose edges is the line AM, which represents in direction and magnitude the magnetic induction B. If the parallelogram is in the plane of the paper, and if AM is drawn upwards from the paper, the volume of the parallelepiped is to be taken positively, or more generally, if the directions of the circuit AB, of the magnetic induction AM, and of the displacement AA', form a right-handed system when taken in this cyclical order. The volume of this parallelepiped represents the increment of the value of p for the secondary circuit due to the displacement of the sliding piece from AB to AB. Electromotive Force acting on the Sliding Piece. 595.] The electromotive force produced in the secondary circuit by the motion of the sliding piece is, by Art. 579, A dp If we suppose AA' to be the displacement in unit of time, then will represent the velocity, and the parallelepiped will represent and therefore, by equation (14), the electromotive force in the negative direction BA. Hence, the electromotive force acting on the sliding piece AB, in consequence of its motion through the magnetic field, is represented by the volume of the parallelepiped, whose edges represent in direction and magnitude-the velocity, the magnetic induction, and the sliding piece itself, and is positive when these three directions are in right-handed cyclical order. Electromagnetic Force acting on the Sliding Piece. 596.] Let i denote the current in the secondary circuit in the positive direction ABC, then the work done by the electromagnetic force on AB while it slides from the position AB to the position A′B′ is (M'—M) i12, where M and M' are the values of M12 in the initial and final positions of AB. But (M'-M), is equal to p'-p, and this is represented by the volume of the parallelepiped on AB, AM, and AA'. Hence, if we draw a line parallel to AB 12 598.] LINES OF MAGNETIC INDUCTION. 219 to represent the quantity AB.i2, the parallelepiped contained by this line, by AM, the magnetic induction, and by AA', the displacement, will represent the work done during this displacement. For a given distance of displacement this will be greatest when the displacement is perpendicular to the parallelogram whose sides are AB and AM. The electromagnetic force is therefore represented by the area of the parallelogram on AB and AM multiplied by 2, and is in the direction of the normal to this parallelogram, drawn so that AB, AM, and the normal are in right-handed cyclical order. Four Definitions of a Line of Magnetic Induction. 597.] If the direction 44', in which the motion of the sliding piece takes place, coincides with AM, the direction of the magnetic induction, the motion of the sliding piece will not call electromotive force into action, whatever be the direction of AB, and if AB carries an electric current there will be no tendency to slide along AA'. Again, if AB, the sliding piece, coincides in direction with AM, the direction of magnetic induction, there will be no electromotive force called into action by any motion of AB, and a current through AB will not cause AB to be acted on by mechanical force. We may therefore define a line of magnetic induction in four different ways. It is a line such that (1) If a conductor be moved along it parallel to itself it will experience no electromotive force. (2) If a conductor carrying a current be free to move along a line of magnetic induction it will experience no tendency to do so. (3) If a linear conductor coincide in direction with a line of magnetic induction, and be moved parallel to itself in any direction, it will experience no electromotive force in the direction of its length. (4) If a linear conductor carrying an electric current coincide in direction with a line of magnetic induction it will not experience. any mechanical force. General Equations of Electromotive Force. 598.] We have seen that E, the electromotive force due to in duction acting on the secondary circuit, is equal to dp where (1) To determine the value of E, let us differentiate the quantity under the integral sign with respect to t, remembering that if the secondary circuit is in motion, x, y, and z are functions of the time. We obtain dH dz 1 dzy ds I dt ds dH dz dx ds dx ds dt Now consider the second term of the integral, and substitute Treating the third and fourth terms in the same way, and col and therefore that the integral, when taken round the closed The terms involving the new quantity are introduced for the sake of giving generality to the expressions for P, Q, R. They disappear from the integral when extended round the closed circuit. The quantity is therefore indeterminate as far as regards the problem now before us, in which the total electromotive force round the circuit is to be determined. We shall find, however, that when we know all the circumstances of the problem, we can assign a definite value to , and that it represents, according to a certain definition, the electric potential at the point x, y, z. The quantity under the integral sign in equation (5) represents the electromotive force acting on the element ds of the circuit. If we denote by TC, the numerical value of the resultant of P, Q, and R, and by e, the angle between the direction of this resultant and that of the element ds, we may write equation (5), The vector is the electromotive force at the moving element ds. Its direction and magnitude depend on the position and motion of ds, and on the variation of the magnetic field, but not on the direction of ds. Hence we may now disregard the circumstance that ds forms part of a circuit, and consider it simply as a portion of a moving body, acted on by the electromotive force E. The electromotive force at a point has already been defined in Art. 68. It is also called the resultant electrical force, being the force which would be experienced by a unit of positive electricity placed at that point. We have now obtained the most general value of this quantity in the case of a body moving in a magnetic field due to a variable electric system. If the body is a conductor, the electromotive force will produce a current; if it is a dielectric, the electromotive force will produce only electric displacement. |