Page:Elementary Text-book of Physics (Anthony, 1897).djvu/378

364 the current due to this electromotive fores will be in the direction opposite to that current which, by its action upon the magnet, would assist the actual motion of the magnet. This current is called an induced current. From the equivalence between a magnetic shell and an electrical current, it is plain that a similar induced current will be produced in a closed circuit by the movement near it of an electrical current or any part of one. Since the joining up or breaking the circuit carrying a current is equivalent to bringing up that same current from an infinite distance, or removing it to an infinite distance, it is further evident that similar induced currents will be produced in a closed circuit when a circuit is made or broken in its presence.

The demonstration of the production of induced currents in § 277 depends upon the assumption that the path of the magnet pole is such that work is done upon it by the current assumed to exist in the circuit. The potential of the magnet pole relative to the current is changed.

The change in potential from one point to another in the magnetic field due to a closed current is (§ 290) $$i (\Omega ' - \Omega + 4 \pi n),$$ and the work done on a magnet pole $$m,$$ in moving it from one point to another, is $$mi (\Omega ' - \Omega + 4 \pi n).$$ In the demonstration of § 277 we may substitute $$m (\Omega ' - \Omega + 4 \pi n)$$ for $$A,$$ and, provided the change in potential be uniform, we obtain at once the expression $$-\frac{m(\Omega ' - \Omega + 4 \pi n)}{t}$$ for the electromotive force due to the movement of the magnet pole. If the change in potential be not uniform, we may conceive the time in which it occurs to be divided into indefinitely small intervals, during any one of which, $$t,$$ it may be considered uniform. Then the limit of the expression $$-\frac{m(\Omega ' - \Omega + 4 \pi n)}{t},$$ as $$t$$ becomes indefinitely small, is the electromotive force during that interval.

The current strength due to this electromotive force is