Page:Elementary Text-book of Physics (Anthony, 1897).djvu/376

362 follows at once a relation between the resistances, expressed in the equation If, therefore, we know the yalue of $$r_{3}$$ and know the ratio of $$r_{1}$$ to $$r_{2},$$ we may obtain the value of $$r_{4}.$$

This method of comparing resistances by means of the Wheatstone's bridge is of great importance in practice. By the use of a form of apparatus known as the British Association bridge the method can be carried to a high degree of accuracy. In this form of the bridge, the portion marked $$ACB$$ (Fig. 87) is a straight cylindrical wire, along which the end of the branch $$CD$$ is moved until a point $$C$$ is found, such that the galvanometer shows no deflection. The two portions of the wire between $$C$$ and $$A,$$ and $$C$$ and $$B,$$ are then the two conductors of which the resistances are $$r_{1}$$ and $$r_{2},$$ and these resistances are proportional to the lengths of those portions (§ 275). The ratio of $$r_{1}$$ to $$r_{2}$$ is therefore the ratio of the lengths of wire on either side of $$C,$$ and only the resistance of $$r_{3}$$ need be known in order to obtain that of $$r_{4}.$$

It is often convenient in determining the relations of current and resistance in a network of conductors to use Ohm's law directly, and consider the difference of potential between the two points on a conductor as equal to the product $$ir.$$ When a part of a circuit is made up of several portions which all meet at two points $$A$$ and $$B,$$ the relation between the whole resistance and that of the separate parts may be obtained easily in this way. For convenience in illustration we will suppose the divided circuit (Fig. 88) made up of only three portions, 1, 2, 3, meeting at the points $$A$$ and $$B,$$ and that no electromotive force exists in those portions. Then the difference of potential between $$A$$ and $$B$$ is $$V_{A} - V_{B} = i_{1}r_{1} = i_{2}r_{2} = i_{3}r_{3}.$$ We have also by Kirchhoff's first law $$-i_{4} = i_{1} + i_{2} + i_{3}.$$ By the combination of these equations we obtain