Wheatstone Bridge

Equivalent forms

R_4 = R_3\frac{R_2}{R_1}

V_g = 0 \text{ at balance}

The reading at balance is exactly zero — a null measurement immune to source voltage and detector calibration.

Symbol	Name	SI unit	Dimension	Range
$R_4$	Unknown resistanceoutput Resistance found at balance	Ω	M·L²·T⁻³·I⁻²	1 – 10000
$R_1$	Resistance 1 First ratio-arm resistor	Ω	M·L²·T⁻³·I⁻²	1 – 10000
$R_2$	Resistance 2 Second ratio-arm resistor	Ω	M·L²·T⁻³·I⁻²	1 – 10000
$R_3$	Resistance 3 Known standard resistor	Ω	M·L²·T⁻³·I⁻²	1 – 10000

Unit systems

Symbol	SI
$R_4$	Ω
$R_1$	Ω
$R_2$	Ω
$R_3$	Ω

Where it holds

Balance condition is exact for DC and linear resistors. Lead and contact resistance limit precision for very small R (use Kelvin/4-wire bridges); for AC and reactive elements use an impedance bridge.

Dimensional analysis

$R_3\cdot (R_2/R_1) = \Omega \cdot (\Omega /\Omega ) = \Omega \checkmark$

Discovery

Samuel Hunter Christie / Charles Wheatstone · 1833

Christie invented the bridge in 1833; Wheatstone popularized and refined it for telegraph testing, so the precision-measurement circuit carries his name despite Christie's priority.

Try this

How do you measure a resistance precisely without trusting your meter — by hunting for the voltage that vanishes.

In a balanced bridge, R₁ = 100 Ω, R₂ = 200 Ω, R₃ = 150 Ω. Find the unknown R₄ that makes the galvanometer read zero.

Research status: stable

Real-world applications

Strain gauges and load cells
RTD and thermistor thermometry
Precision resistance metrology
Pressure and force sensors

Common misconceptions

You must know the supply voltage — at balance the result depends only on the resistor ratio
The detector must be calibrated — you only need it to detect zero, not measure a value

Experimental verification

Adjust a calibrated

R_{3}

until the galvanometer nulls; the computed

R_{4}

matches a precision standard to ppm levels, independent of supply voltage — the hallmark of a null method.

Derivation

The bridge is two dividers across the same supply.

The galvanometer node voltages are

V\cdot R_{2}/(R_{1}+R_{2})

and

V\cdot R_{4}/(R_{3}+R_{4})

.

Setting them equal (zero detector current) gives

R_{2}/(R_{1}+R_{2}) = R_{4}/(R_{3}+R_{4})

, which rearranges to

R_{1}/R_{2} = R_{3}/R_{4}

, i.e.

R_{4} = R_{3}R_{2}/R_{1}

.

Limiting cases

R_1 = R_2

⟶

R_4 = R_3

Equal ratio arms make the bridge a direct comparator: unknown equals standard.

R_2 ≫ R_1⟶ R_4 ≫ R_3A large ratio arm multiplies the standard, extending the measurable range.

What if…

What if the unknown is a strain gauge?

Strain changes its resistance, unbalancing the bridge; the tiny output voltage is the basis of load cells and scales.

What if you swap the galvanometer for an AC detector and use impedances?

You get an impedance bridge (Wien, Maxwell) that measures capacitance and inductance the same way.

1

Finding the unknown arm

Given ·

R 1:: 100
R 2:: 200
R 3:: 150

Find · R_4 at balance

Steps

Balance: $R_1/R_2 = R_3/R_4$
$R_4 = R_3\cdot R_2/R_1$
$R_4 = 150\times (200/100) = 300 \Omega$

Result ·

R_4 = 300 \Omega

nulls the galvanometer.

resistors series parallelKirchhoff's Voltage Law→Thévenin's Theorem→resistivity