**9.3. Measurement of Resistance by the Wheatstone Bridge**. In Wheatstone bridge

method of measuring resistances, a resistor of unknown resistance is balanced against resistors of known resistances. Though based on comparison of resistances, this is one of the most accurate methods of measuring resistances as it is independent of the calibration of the indicating instrument and relies upon .the null-point method.

The four branches of the network ABCDA (see Fig. 33) have two known resistances P and Q, a known variable resistance R and the unknown resistance X. A battery E is connected through a switch S_{1} to junctions A and C; and a galvanometer G, a variable resistor Z and a switch S_{2} are in series across Band D. The function of Z is merely to protect G against an excessive current should the system be seriously out of balance when S_{2} is closed.

After closing S_{1} and S_{2}; R is adjusted until there is no deflection on G with the resistance of Z reduced to zero. Junctions Band D are then at the same potential,

so that p.d. between A and B is the same as that between A and D, and the p.d. between B and C is the same as that between C and D.

Fig. 33. Wheatstone bridge.

Let I_{1} and I_{2} be the currents through P and R respectively when the bridge is balanced. From Kirchhoff’s first law it follows that since there is no current through G, the currents through Q and X are also I_{1} and I_{2} respectively.

But potential difference (p.d.) across P = PI_{1}

Potential differences across R = RI_{2}

PI_{1} =RI_{2} … (12)

Also potential difference across Q = QI_{1}

Potential difference across X = XI_{2}

QI_{1} =XI_{2} … (13)

Dividing (13) by (12), we get

which is the relation of Wheatstone bridge. P and Q are usually called the ratio arms and X as the balance or rheostat arm. The battery and the galvanometer can be interchanged without affecting the relation.

**Note**. The resistances P and Q may take the form of the resistance of a slide-wire, in which case R may be a fixed value and the balance is obtained by moving a sliding contact along the wire. If the wire is homogeneous and of uniform section, ‘the ratio of P to Q is the same as the ratio of the lengths of wire in the respective arms.

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