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Wheatstone bridge derivation

Kirchhoff's loop rule on the two loops through the galvanometer, with no current in it, gives P/Q = R/S. If the bridge is off balance the same laws give the galvanometer current: with S = 40 Ω instead of 30 Ω it is 0.0161 A, from D to B.

P 10 Ω0.19 AQ 20 Ω0.21 AR 15 Ω0.12 AS 40 Ω0.1 AG 5 Ω0.0161 A+6 VABCD
S is a little too big, so a small current runs from D to B through G.

Wheatstone bridge

P from A to B, Q from B to C, R from A to D, S from D to C, the galvanometer G from B to D. It balances when P/Q = R/S; then no current flows through G.

Galvanometer current0.0161 Afrom D to B
S for balance30 Ω
Cell current0.31 A
  1. P/Q = 10 / 20 = 0.5 and R/S = 15 / 40 = 0.375.
  2. The ratios differ, so the bridge is not balanced. It would balance at S = QR/P = 20 × 15 / 10 = 30 Ω.
  3. Kirchhoff's laws at B and D give VB = 4.11 V and VD = 4.19 V above C.
  4. Ig = (VB - VD) / G = (4.11 - 4.19) / 5 = 12/745 ≈ 0.0161 A, from D to B.
  5. The resistance between A and C works out to 1490/77 ≈ 19.35 Ω.
  6. Cell current: I = E / R = 6 / 19.35 = 231/745 ≈ 0.31 A.
Derivation

Why it works

  1. Let I₁ flow through P and I₂ through R from A. With a current Ig through the galvanometer from B to D, Q carries I₁ - Ig and S carries I₂ + Ig.
  2. Loop ABDA: I₁P + IgG - I₂R = 0.
  3. Loop BCDB: (I₁ - Ig)Q - (I₂ + Ig)S - IgG = 0.
  4. At balance Ig = 0, so the two loops give I₁P = I₂R and I₁Q = I₂S.
  5. Dividing the first by the second: P/Q = R/S. It does not depend on the cell or on G.
Questions

Wheatstone bridge derivation: quick answers

How is the Wheatstone bridge condition derived?

Write the loop rule for ABDA and BCDB with the galvanometer current set to zero. That gives I₁P = I₂R and I₁Q = I₂S, and dividing them gives P/Q = R/S.

What happens when the bridge is not balanced?

A current flows through the galvanometer from the higher of B and D to the lower. Here, with S = 40 Ω, it is 0.0161 A from D to B.

Does the balance depend on the emf of the cell?

No. The condition P/Q = R/S has no E or G in it, so changing the cell or the galvanometer does not move the balance point.

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