Page:Elementary Text-book of Physics (Anthony, 1897).djvu/379

§ 306] If the induced current be steady, the total quantity of electricity flowing in the circuit is expressed by $$i't = -\frac{m(\Omega ' - \Omega + 4 \pi n)}{r}\cdot$$ The total quantity of electricity flowing in the circuit depends, therefore, only upon the initial and final positions of the magnet pole, and the number of times it passes through the circuit, and not upon its rate of motion. The electromotive force due to the movement of the magnet, and consequently the current strength, depends, on the other hand, upon the rate at which the potential changes with respect to time.

A more general statement of the mode in which induced currents are produced may be given in terms of the changes in the number of tubes of induction which pass through the circuit. When the number of tubes of induction which pass through a circuit is altered, an electromotive force is induced in the circuit which is proportional to the rate of change of the number of tabes of induction. This law may be easily proved, as in the special case already considered, if the change in the number of tubes of induction be produced by a movement of magnet poles or their equivalents, and not by changes in other currents in the field; in case there are other currents in the field, the interactions between them introduces conditions which cannot be discussed by elementary methods. The law, however, is a perfectly general one, and holds for all cases in which the tubes of induction passing through the circuit change in number.

While we cannot, by elementary methods, determine exactly the laws of the production of an induced current in a circuit by changes in the currents in neighboring circuits, we may yet form some idea of the induced current by considering the magnetic field about the circuits. Suppose that a current traverses circuit 1 and that there is no current in circuit 2; circuit 2 encloses a number of tubes of induction due to the current in circuit 1. If the current in circuit 1 be suddenly interrupted, these tubes of induction are removed from circuit 2, and from the dynamical principle that a change is resisted by the non-conservative forces to which it gives rise, there will arise in circuit 2 a current tending to maintain the tubes;