Physics · Ch 2 — Current Electricity

Wheatstone's Bridge

2.5.3

Wheatstone's Bridge

An important, extremely common practical application of Kirchhoff's rules is the Wheatstone's bridge, used both to compare resistances and to determine an unknown resistance in a network. The bridge (Figure 2.25) consists of four resistances P, Q, R and S arranged in a diamond shape connecting four nodes A, B, C, D: P between A and B, Q between B and C, R between A and D, and S between D and C. A battery is connected across the A-C diagonal (driving a total current I), and a sensitive galvanometer G (of resistance G, carrying current $I_G$ ) is connected across the B-D diagonal.

Applying Kirchhoff's current rule at junctions B and D gives $I_1-I_G-I_3=0$ (2.45) and $I_2+I_G-I_4=0$ (2.46). Applying Kirchhoff's voltage rule to loop ABDA gives $I_1P+I_GG-I_2R=0$ (2.47), and to loop ABCDA gives $I_1P+I_3Q-I_4S-I_2R=0$ (2.48). The bridge is said to be balanced exactly when points B and D sit at the same potential, so that no current at all flows through the galvanometer ( $I_G=0$ ). Substituting $I_G=0$ into equations (2.45)-(2.47) gives $I_1=I_3$ (2.49), $I_2=I_4$ (2.50), and $I_1P=I_2R$ (2.51); using (2.51) in (2.48) then gives $I_3Q=I_4S$ (2.52). Dividing equation (2.52) by equation (2.51) gives the elegant balance condition:

$\dfrac{P}{Q} = \dfrac{R}{S} \qquad (2.53)$ …

Figure 2.25Wheatstone's bridge

What this figure shows. A diamond-shaped network of four resistors P, Q, R, S connecting four nodes A, B, C, D: P between A and B, Q between B and C, R between A and D, and S between D and C. A battery of emf $\varepsilon$ (driving total current I) is connected across the A-C diagonal, and a galvanometer G (carrying current $I_G$ ) is connected across the B-D diagonal. Branch currents $I_1$ through P, $I_2$ through R, $I_3$ through Q and $I_4$ through S are all labelled, feeding directly into the Kir …

Misc ~note-galvanometerWhat a galvanometer is used for

Worked out. A short aside defining the galvanometer: it is an instrument used to detect and measure even very small electric currents, and it is extensively used, as in the Wheatstone's bridge, to compare the potential difference between different parts of a circuit. …

Misc Example 2.23Finding the unknown fourth arm of a balanced bridge

Worked out. In a balanced Wheatstone's bridge, $P=100\ \Omega$ , $Q=1000\ \Omega$ and $R=40\ \Omega$ (the galvanometer shows zero deflection); the value of S is required. Using the balance condition $P/Q=R/S$ , rearranged as $S=(Q/P)\times R = (1000/100)\times40 = 10\times40 = 400\ \Omega$ . …

Misc Example 2.24Finding an unknown component embedded in one arm of a balanced bridge

Worked out. In a balanced Wheatstone's network, $P=500\ \Omega$ , $Q=800\ \Omega$ , $S=1000\ \Omega$ and the fourth arm is $R=x+400\ \Omega$ ; the value of x is required. Using the balance condition $P/Q=R/S$ : $500/800=(x+400)/1000$ , so $x+400=1000\times(5/8)=625$ , giving $x=625-400=225\ \Omega$ . …

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