9.17 Wheatstone Bridge

9.17 Wheatstone Bridge

It is an electric circuit. The Wheatstone bridge circuit shown in Fig. 9.31 consists of four resistances $R_{1}$, $R_{2}$, $R_{3}$, and $R_{4}$ connected in such a way so as to form a mesh ABCDA. A battery is connected between points $A$ and $C$. A sensitive galvanometer of resistance $R_{g}$ is connected between points $B$ and $D$. If the switch $S$ is closed, a current will flow through the galvanometer. We have to determine the condition under which no current flows through the galvanometer even after the switch is closed. For this purpose, we analyse this circuit using Kirchhoff's second rule. We consider two loops ABDA and BCDB and assume anticlockwise loop currents $I_{1}$ and $I_{2}$ through the loops respectively. The Kirchhoff's second rule as applied to loop ABDA gives:

Similarly, by applying the Kirchhoff's second rule to loop BCDB, we have:

The current flowing through the galvanometer will be zero if, $I_{1} - I_{2} = 0$ or $I_{1} = I_{2}$. With this condition Eq. 9.49 and Eq. 9.50 reduce to:

and

Dividing Eq. 9.51 by Eq. 9.52, we have:

As $I_{1} = I_{2}$, therefore:

Thus, whenever the condition of Eq. 9.54 is satisfied, no current flows through the galvanometer and it shows no deflection, or conversely when the galvanometer in the Wheatstone bridge circuit shows no deflection, Eq. 9.54 is satisfied. If we connect three resistances $R_{1}$, $R_{2}$ and $R_{3}$ of known adjustable values and a fourth resistance $R_{4}$ of unknown value and the resistances $R_{1}$, $R_{2}$ and $R_{3}$ are so adjusted that the galvanometer shows no deflection, then from the known resistances $R_{1}$, $R_{2}$ and $R_{3}$, the unknown resistance $R_{4}$ can be determined by using Eq. 9.54.