An important application of Kirchoff’s law is the Wheatstone’s bridge (Figure).

Wheatstone’s network consists of resistances P, Q, R and S connected to form a closed path. A cell of emf E is connected between points A and C. The current I from the cell is divided into I_{1}, I_{2}, I_{3} and I_{4} across the four branches. The current through the galvanometer is Ig. The resistance of galvanometer is G.

**Fig: Wheatstone’s bridge**

Applying Kirchoff’s current law to junction B,

**I _{1} – I_{g} – I_{3} = 0 … (1)**

Applying Kirchoff’s current law to junction D

**I _{2} + I_{g} – I_{4} = 0 … (2)**

Applying Kirchoff’s voltage law to closed path ABDA

**I _{1} P + I_{g} G – I_{2} R = 0 … (3)**

Applying Kirchoff’s voltage law to closed path ABCDA

**I _{1} P + I_{3} Q – I_{4} S – I_{2} R = 0 … (4)**

When the galvanometer shows zero deflection, the points B and D are at same potential and I_{g} = 0. Substituting I_{g} = 0 in equation (1), (2) and (3)

I_{1} = I_{3} … (5)

I_{2} = I_{4} … (6)

I_{1} P = I_{2} R … (7)

Substituting the values of (5) and (6) in equation (4)

I_{1} P + I_{1} Q – I_{2} S – I_{2} R = 0

**I _{1} (P + Q) = I_{2} (R+S) … (8)**

Dividing (8) by (7)

I_{1} (P + Q) / I_{1} P = I_{2} (R+S) / I_{2} R

(P + Q) / P = I (R+S) / R

1 + Q/P = 1 + S/R

so, **Q/P = S/R** or, **P/Q = R/S**

This is the condition for bridge balance. If P, Q and R are known, the resistance S can be calculated.

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