Page:Elementary Text-book of Physics (Anthony, 1897).djvu/377

§ 306] The current in the divided circuit equals the difference of potential between $$A$$ and $$B$$ multiplied by the sum of the reciprocals of the resistances of the separate portions. If we set this sum equal to $$\frac{1}{r},$$ and call $$r$$ the resistance of the divided circuit, we may say that the reciprocal of the resistance of a divided circuit is equal to the sum of the reciprocals of the resistances of the separate portions of the circuit. When there are only two portions into which the circuit is divided, one of them is usually called a shunt, and the circuit a shunt circuit.

The rules for joinijig up sets of voltaic cells in circuits so as to accomplish any desired purpose may be discussed by the same method. Let us suppose that there are $$n$$ cells, each with an electromotive force $$e$$ and an internal resistance $$r,$$ and that the external resistance of the circuit is $$s.$$ If $$m$$ be a factor of $$n,$$ and if we join up the cells with the external resistance so as to form a divided circuit of $$m$$ parallel branches, each containing $$\frac{n}{m}$$ cells, we shall have for the electromotive force in such a circuit $$\frac{ne}{m},$$ and for the resistance of the circuit $$s + \frac{nr}{m^2}\cdot$$ The current in the circuit is therefore $$i = \frac{mne}{m^2 s + nr}\cdot$$ Two cases may arise which are common in practice. The resistance $$s$$ of the external circuit may be so great that, in comparison with $$m^2 s, \, nr$$ may be neglected. In that case $$i$$ is a maximum when $$m = 1,$$ that is, when the cells are arranged tandem, or in series, with their unlike poles connected. On the other hand, if $$m^2 s$$ be very small as compared with $$nr,$$ it may be neglected, and $$i$$ becomes a maximum when $$m = n,$$ that is, when the cells are arranged abreast, or in multiple arc, with their like poles in contact.

306. Induced Currents.—It was shown in § 277 that the movement of a magnet in the neighborhood of a closed circuit will give rise, in general, to an electromotive force in the circuit, and that