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 S1 to junctions A and C; and a galvanometer G, a variable resistor Z and a switch S2 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 S2 is closed.
After closing S1 and S2; 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 I1 and I2 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 I1 and I2 respectively.
But potential difference (p.d.) across P = PI1
Potential differences across R = RI2
PI1 =RI2 … (12)
Also potential difference across Q = QI1
Potential difference across X = XI2
QI1 =XI2 … (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.