Wheatstone Bridge Principle

A Wheatstone bridge is an electrical circuit used to measure an unknown resistance with exceptional precision. It consists of four resistors arranged in a diamond (rhombus) shape, with a battery connected across one diagonal and a sensitive galvanometer (a current detector) connected across the other. The beauty of the device lies in the fact that, at the 'balanced' condition, the galvanometer reads exactly zero — meaning you do not actually need to measure any current at all. Instead, you exploit the absence of current to deduce an unknown resistance from three known values. Think of a Wheatstone bridge like a perfectly balanced set of scales. When both pans hold identical weights, the beam is horizontal and you know the unknown mass equals the known mass. Similarly, when the bridge is balanced, the ratio of resistances on one side of the diamond exactly matches the ratio on the other side. This null-detection method is far more accurate than directly measuring voltage or current, because it eliminates errors due to the internal resistance of the measuring instruments themselves. The principle was first described by Samuel Hunter Christie in 1833 and later popularised by Sir Charles Wheatstone in 1843. Today, Wheatstone bridges remain fundamental in sensor technology — strain gauges, thermistors, and pressure transducers all use bridge circuits to convert tiny physical changes into measurable electrical signals.

Key takeaways
  • The Circuit Layout
  • Current Paths Without Balance
  • The Balanced Condition
  • Deriving the Balance Equation Formally
  • Sensitivity and Practical Considerations

The Circuit Layout

Label the four resistors P, Q, R, and X arranged in a diamond. Node A is the top junction, node B is the bottom, node C is the left, and node D is the right. The battery (EMF ε) connects between nodes A and B. The galvanometer connects between nodes C and D. Resistor P sits between A and C, Q between A and D, R between C and B, and X (the unknown) between D and B.

Current Paths Without Balance

When the bridge is unbalanced, current flows through both branches: one branch carries current through P then R, the other through Q then X. Because the resistances are unequal, the potential at node C differs from the potential at node D, so current flows through the galvanometer. The galvanometer deflects left or right depending on which node is at higher potential.

The Balanced Condition

Balance is achieved by adjusting a known variable resistor until the galvanometer reads zero (no deflection). At this moment, nodes C and D are at identical electric potential. No current flows through the galvanometer, so we can treat each branch as a simple series circuit. The potential divider rule gives: V_AC / V_CB = P/R and V_AD / V_DB = Q/X. Setting these equal (since V_C = V_D) yields the balance equation: P/Q = R/X.

Deriving the Balance Equation Formally

At balance, I_g = 0, so current I₁ flows through P and R in series, and current I₂ flows through Q and X in series. The voltage across P equals the voltage across Q (both measured from node A): I₁P = I₂Q. The voltage across R equals the voltage across X (both measured to node B): I₁R = I₂X. Dividing the first equation by the second: P/R = Q/X, which rearranges to P·X = Q·R, giving X = QR/P. This lets you calculate the unknown resistance X precisely.

Sensitivity and Practical Considerations

Bridge sensitivity describes how large a galvanometer deflection results from a small imbalance. A highly sensitive galvanometer finds the null point more accurately. In practice, the battery should have low internal resistance and the galvanometer should be connected via a protective resistor initially (to prevent damage from large currents when far from balance). Resistors P and Q are often a fixed ratio arm (e.g. 1:1, 1:10, or 10:1), while R is a calibrated variable resistor called the rheostat arm.

The Key Equation

PQ=RX⟹X=Q⋅RP\frac{P}{Q} = \frac{R}{X} \quad \Longrightarrow \quad X = \frac{Q \cdot R}{P}
  • P — Ratio arm resistor (Ω (ohm)): Known resistor forming one arm of the ratio; its value relative to Q sets the measurement range.
  • Q — Ratio arm resistor (Ω (ohm)): Known resistor forming the other arm of the ratio; P and Q together define the multiplying factor.
  • R — Variable (rheostat) arm (Ω (ohm)): Calibrated variable resistor adjusted until the galvanometer reads zero; its value at balance is read off a dial.
  • X — Unknown resistance (Ω (ohm)): The resistance being measured; calculated from the other three known values once balance is achieved.

Worked Example

Problem. In a Wheatstone bridge, P = 100 Ω and Q = 200 Ω (ratio arm). The variable resistor R is adjusted until the galvanometer shows no deflection; at that point R = 350 Ω. Calculate the unknown resistance X.

X=Q×RP=200×350100=700 ΩX = \frac{Q \times R}{P} = \frac{200 \times 350}{100} = 700 \ \Omega

Answer. The unknown resistance X = 700 Ω

Figure
Diagram
Wheatstone Bridge CircuitACDBPQRX=?GGalvanometer+−Battery(EMF ε)Balance condition: P/Q = R/X → X = QR/P
Origins · 1 minSir Charles Wheatstone and how this was discovered
SC
Sir Charles Wheatstone
1802–1875 · British

Wheatstone popularised the bridge circuit in his 1843 paper to the Royal Society, demonstrating it as a precise method for resistance measurement. Although Samuel Hunter Christie invented the circuit in 1833, Wheatstone's thorough analysis, improvements, and promotion of the technique led to it bearing his name. He also made major contributions to telegraphy, the stereoscope, and the concertina. Wheatstone was so shy about public speaking that his friend Michael Faraday had to deliver Wheatstone's own Royal Institution lecture when Wheatstone lost his nerve and fled the building just before going on stage.

  • Ohm's Law — Ohm's Law (V = IR) underpins the derivation of the balance condition; each resistor in the bridge obeys it to establish the potential at each node.
  • Resistors in Series and Parallel — The bridge arms act as series pairs; understanding series and parallel combinations is essential to analysing current flow when the bridge is unbalanced.
  • Potential Divider — The Wheatstone bridge is effectively two potential dividers in parallel; the balance condition arises when both dividers produce the same output voltage.
  • Galvanometer and Ammeter — The galvanometer is the null detector at the heart of the bridge; knowing its sensitivity and internal resistance is crucial for accurate measurements.