Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/170

is turned from the direction of the current $i\,ds$ to that of the magnetic induction ${\mathfrak {B}}$.
Fig. 22. We may express in the language of Quaternions, both the direction and the magnitude of this force by saying that it is the vector part of the result of multiplying the vector $i\,ds$, the element of the current, by the vector ${\mathfrak {B}}$, the magnetic induction.

491.] We have thus completely determined the force which acts on any portion of an electric circuit placed in a magnetic field. If the circuit is moved in any way so that, after assuming various forms and positions, it returns to its original place, the strength of the current remaining constant during the motion, the whole amount of work done by the electromagnetic forces will be zero. Since this is true of any cycle of motions of the circuit, it follows that it is impossible to maintain by electromagnetic forces a motion of continuous rotation in any part of a linear circuit of constant strength against the resistance of friction, &c.

It is possible, however, to produce continuous rotation provided that at some part of the course of the electric current it passes from one conductor to another which slides or glides over it.

Fig. 23. When in a circuit there is sliding contact of a conductor over the surface of a smooth solid or a fluid, the circuit can no longer be considered as a single linear circuit of constant strength, but must be regarded as a system of two or of some greater number of circuits of variable strength, the current being so distributed among them that those for which $N$ is increasing have currents in the positive direction, while those for which $N$ is diminishing have currents in the negative direction.

Thus, in the apparatus represented in Fig. 23, $OP$ is a moveable conductor, one end of which rests in a cup of mercury $O$, while the other dips into a circular trough of mercury concentric with $O$.