Wheatstone Bridge

Walk into the undergraduate electricity lab at IISc Bangalore on a Thursday morning. You will find rows of wooden tables, each carrying a brass-and-bakelite "post office box" — four decade resistance dials, a small galvanometer with a mirror-reflecting needle, a battery of Leclanché cells, and two brass keys used to close the battery and galvanometer circuits one after the other. The student's task is to measure the resistance of an unknown piece of nichrome wire, clamped between two terminals. The procedure is memorised, generations deep: set the ratio dials to 10:10, adjust the decade resistor until the galvanometer settles at zero, read the decade's value, done. The unknown's resistance is now known to four significant figures.

The physics in that box is the Wheatstone bridge, a circuit that exploits a symmetry trick to turn resistance measurement into a null measurement — find the setting at which nothing happens. Null measurements are among the most accurate techniques in all of experimental physics: they don't depend on calibration of a pointer, don't depend on the voltage of the battery, don't depend on the galvanometer's sensitivity beyond its ability to register "zero". The precision is limited only by how finely you can subdivide the known resistors.

The same circuit, wired into a modern strain gauge, sits under every Indian Railways weighbridge you pass on the road to Nashik or Gwalior. It sits inside the piezoelectric accelerometer mounted on a vibrating concrete slab in the IIT-Madras structural-engineering lab. It is on the printed circuit board inside every electronic kitchen scale in Indian households. One diamond, four resistors, one detector — and any physical quantity that changes resistance (strain, temperature, humidity, force, pressure) can be measured with a bridge.

This chapter derives why.

The circuit — four resistors in a diamond

The classical Wheatstone bridge is drawn as a diamond with four vertices A, B, C, D. A battery (EMF \varepsilon, internal resistance r) drives current between A and C — the opposite pair of vertices. The galvanometer G is connected between B and D — the other opposite pair.

The resistors P, Q, R, S are labelled following the convention used in Indian physics-lab handbooks:

• P between A and B (top-right arm, often called the ratio arm)
• Q between B and C (bottom-right arm, the other ratio arm)
• R between A and D (top-left arm, the standard or known resistor, sometimes labelled R_{s})
• S between D and C (bottom-left arm, traditionally the unknown — hence the symbol S, which unfortunately is the same letter used for the "standard" in some newer textbooks; pay attention to which is which)

Some textbooks swap which pair is the "ratio arm" and which is the "unknown arm"; the physics is symmetric under that swap, and the balance condition takes exactly the same form.

Deriving the balance condition

The central claim: The bridge is balanced — meaning no current flows through the galvanometer — exactly when P/Q = R/S.

The proof comes from writing down what "no galvanometer current" means, applying Kirchhoff's laws, and reading off the constraint.

Assumptions. The four resistors are pure ohmic conductors (constant resistance, not temperature- or current-dependent). The battery has EMF \varepsilon and internal resistance r; neither of these will appear in the final balance condition, so they don't need to be known. The galvanometer has its own resistance R_{g}, which likewise will not appear in the balance condition.

Step 1. Assume, as a hypothesis, that I_{g} = 0. I will show this is consistent with P/Q = R/S.

If no current flows through the galvanometer, then by Kirchhoff's current law at junction B, the current arriving at B through P must leave B through Q. Similarly at junction D: current arriving through R must leave through S. So

Why: the galvanometer is a dead-end at B and D when I_{g} = 0, so by KCL the current through the right arm (P then Q) must be single-valued, and similarly for the left arm.

Step 2. The galvanometer reading I_{g} depends on the potential difference between B and D.

If V_{B} \neq V_{D}, a current will flow through G. For I_{g} to be exactly zero, we need V_{B} = V_{D}.

Let V_{A} be the potential at vertex A and V_{C} at vertex C. The potential drop from A to B (through resistor P, in the direction of current I_\text{right}) is I_\text{right} P:

Similarly, going from A down to D through resistor R:

Setting V_{B} = V_{D} gives:

Why: the balance condition says the IR drops along the two top arms are equal, so both junctions B and D reach the same potential below A. The V_{A} cancels because it is common to both sides.

Step 3. Do the same computation going from C up to B and from C up to D.

V_{B} = V_{C} + I_\text{right} Q (going backwards up the right arm raises the potential).

V_{D} = V_{C} + I_\text{left} S.

Setting V_{B} = V_{D} again:

Step 4. Divide equation (4) by equation (5).

The currents cancel cleanly:

Why: dividing (4) by (5) is the crucial move. The currents I_\text{right} and I_\text{left} — quantities that depend on the battery EMF, the battery's internal resistance, and on each resistor's value — cancel exactly in the ratio. What is left is a relation among the four resistances alone. This is what makes the bridge measurement EMF-independent and galvanometer-resistance-independent: those quantities don't appear in (6) because they divided out.

Step 5. Verify the logic is two-way.

I assumed I_{g} = 0 and derived P/Q = R/S. For the measurement to work, I also need the reverse: given that P/Q = R/S, it must follow that I_{g} = 0.

That direction is immediate: if P/Q = R/S, then with the currents dividing freely (some I_\text{right} through the right arm and some I_\text{left} through the left arm), one can check directly that V_{B} = V_{D}, so no current is driven through G. The galvanometer sees equal potentials on its two ends; Ohm's law says no current flows. Balance is both necessary and sufficient.

Why the balance is independent of \varepsilon, r, and R_{g}

Equation (6) mentions only P, Q, R, S. It does not mention the battery EMF \varepsilon or the internal resistance r, nor the galvanometer resistance R_{g}. This is a feature, not a coincidence. It means:

1. You can swap a weaker battery for a stronger one. The balance point does not move.
2. You can swap a cheap galvanometer for a sensitive one. The balance point does not move.
3. You can swap in a digital multimeter set to μA as the detector. Same balance point.

The only thing that matters is that the detector can resolve "zero" — identify the setting of the adjustable resistor where I_{g} is below the detector's noise floor. A more sensitive detector lets you make P, Q, R more precise, because you can read out a smaller I_{g} as "zero". But the value of the balance setting does not depend on the detector.

This is the trademark of a null measurement. You don't have to calibrate the detector's scale, just trust that it can recognise zero.

Interactive — watching the galvanometer deflect

The curve shown is approximate, assuming series-parallel reduction. The exact form for an unbalanced bridge comes from Thevenin's theorem (see going-deeper), but the zero crossing is rigorously at the balance condition, independent of the detailed curve shape.

Measuring an unknown resistance — the classical procedure

Suppose you want to measure S, a piece of constantan wire salvaged from an old decade box. You have:

• Two precision resistors of known value P = 100.00 Ω and Q = 100.00 Ω (the ratio arm, set 1:1).
• A decade resistance box for R, adjustable from 0 to 1000 Ω in steps of 0.1 Ω.
• A sensitive galvanometer.
• A battery.

Procedure.

Step 1. Wire up the bridge with S in its place and R set to your best guess.

Step 2. Briefly close the battery key, then briefly close the galvanometer key, and observe the deflection. The galvanometer deflects to one side. Note the sign.

Step 3. Adjust R upward or downward. If the deflection was to the right, try increasing R; observe which way the deflection moves. A few trial settings will give you the direction.

Step 4. Binary-search to zero deflection. If R = 75.0 Ω gives a small left deflection and R = 76.0 Ω gives a small right deflection, try 75.5, then 75.3, etc. Within a few minutes, the deflection is null — call this setting R_{0}.

Step 5. Read off the unknown:

With P = Q, S = R_{0}. With P = 10, Q = 1000, S = R_{0}/100 — a ratio arm of 1:100 extends the bridge's range ten-thousand-fold downward. With P = 1000, Q = 10, S = 100 R_{0} — the same range extended upward.

The ratio-arm trick is what makes a single decade box of R (0 to 1000 Ω) measure everything from 10^{-2} Ω (0.01 Ω, about the internal resistance of a car battery) to 10^{5} Ω (100 kΩ, a typical pull-up resistor in a microcontroller circuit) without changing the hardware.

Worked examples

Common confusions

• "The bridge only works at balance." The bridge is most precise at balance (null reading = no dependence on battery, galvanometer). But it is perfectly usable unbalanced — that is exactly the strain-gauge mode. In strain-gauge applications, you deliberately operate slightly off balance so the small deflection is proportional to the small physical change you are trying to measure.

• "The galvanometer's resistance R_{g} affects the balance point." It does not. R_{g} appears in the unbalanced galvanometer current (through Thevenin's R_{th}+R_{g}), but not in the zero-crossing setting. The balance condition P/Q = R/S is a pure statement about the four ratio arms.

• "A more sensitive galvanometer gives a different balance." No — it gives a more precise balance, by reducing the deflection threshold you can resolve as "zero". The numerical value of the balance remains P/Q = R/S.

• "If I put the battery between B and D and the galvanometer between A and C, it won't work." It will — the bridge is symmetric under swapping the battery and galvanometer. The balance condition is still P/Q = R/S, and the null measurement still works. Some old British designs used this swapped topology. The derivation in this chapter picked one arrangement for concreteness.

• "The unbalanced galvanometer current is given by Ohm's law across the galvanometer with V_{th}." Almost — you also need to add the Thevenin resistance in series: I_{g} = V_{th}/(R_{th}+R_{g}). Forgetting R_{th} gives an overestimate by a factor of 3-5 for typical bridge geometries.

• "Balance condition holds only for DC." It holds for AC too, with resistances replaced by complex impedances: Z_{P}/Z_{Q} = Z_{R}/Z_{S}. This is how AC bridges — the Wien bridge, the Maxwell bridge, the Hay bridge — measure capacitance and inductance. They are 20th-century extensions of Wheatstone's 1843 idea.

• "The ratio P/Q must be small for the bridge to be sensitive." Not quite. The bridge's range is extended by large P/Q ratios (to measure very small or very large S), and its sensitivity (galvanometer current per unit deviation of S) is highest when P + Q \approx R + S. Good bridge designs balance range, sensitivity, and current draw.

Where this leads next

• Meter Bridge and Potentiometer — the practical form of the Wheatstone bridge used in every Indian physics lab, where two of the four arms are replaced by a uniform resistance wire and a sliding contact.
• Kirchhoff's Laws — the general circuit-analysis rules from which the balance condition was derived.
• Combinations of Resistors — the series and parallel rules that the bridge uses implicitly when it decomposes into two clean arms at balance.
• EMF and Internal Resistance — why the Wheatstone bridge is robust against changes in battery voltage and internal resistance (they cancel in the balance condition).
• AC Circuits and Impedance — the generalisation of the bridge concept to alternating currents, leading to the Maxwell, Wien, and Hay bridges.
• Galvanometers and Ammeters — the detector that sits at the heart of the bridge, and how its sensitivity is optimised for null measurements.