Wheatstone Bridge Formula
The Wheatstone bridge measures unknown resistance with high precision.
Learn the balance condition formula with practical examples.
The Formula
A Wheatstone bridge is an electrical circuit used to measure unknown resistance with great accuracy. It consists of four resistors arranged in a diamond shape, with a voltage source across one diagonal and a galvanometer (sensitive current meter) across the other.
The bridge is "balanced" when no current flows through the galvanometer. At balance, the ratio of resistors in one branch equals the ratio in the other: R₁/R₂ = R₃/R_x. Rearranging gives the unknown resistance R_x.
The Wheatstone bridge was invented by Samuel Hunter Christie in 1833 and later popularized by Sir Charles Wheatstone in 1843. It remains widely used in strain gauges, temperature sensors, and precision measurement instruments. Modern variations include the Kelvin bridge (for very low resistances) and the Wien bridge (for AC measurements).
Variables
| Symbol | Meaning |
|---|---|
| R_x | Unknown resistance to be measured (in ohms, Ω) |
| R₁ | Known resistance in the first branch (Ω) |
| R₂ | Known resistance in the first branch, paired with R₁ (Ω) |
| R₃ | Known adjustable resistance in the second branch (Ω) |
Example 1
A Wheatstone bridge is balanced when R₁ = 100 Ω, R₂ = 400 Ω, and R₃ = 50 Ω. What is R_x?
Apply the balance condition: R_x = R₃ × (R₂ / R₁)
R_x = 50 × (400 / 100) = 50 × 4
R_x = 200 Ω
Example 2
A strain gauge Wheatstone bridge has R₁ = R₂ = R₃ = 120 Ω when unstrained. The strain gauge (R_x) changes to 120.5 Ω under load. What is the ratio R_x/R₃?
At balance: R₁/R₂ = R₃/R_x, so R_x/R₃ = R₂/R₁
Without strain: R_x = 120 Ω, ratio = 120/120 = 1.000 (balanced)
With strain: R_x = 120.5 Ω, ratio = 120.5/120 = 1.00417
The bridge is now unbalanced. The galvanometer detects a voltage proportional to the 0.417% change.
The change of 0.5 Ω (0.42%) is detected as an imbalance voltage
When to Use It
The Wheatstone bridge is used for precise resistance measurements.
- Strain gauge measurements in structural engineering
- Temperature measurement with resistance thermometers (RTDs)
- Precision resistance calibration in laboratories
- Sensor circuits that detect small resistance changes