Wheatstone Bridge: Formula, Derivation and Solved Examples
What Is a Wheatstone Bridge?
A Wheatstone bridge is an electrical circuit used to find an unknown resistance by comparing it with known resistances. Four resistors are connected to form a bridge, a battery supplies the circuit, and a galvanometer checks whether current flows through the middle branch.
Think of it like a balancing scale: when the resistance ratios on the two sides match, the bridge is balanced. The galvanometer then shows zero current. This is called the balance point.
The circuit has four resistive arms. A power supply is connected across one diagonal, while a galvanometer connects the other two junctions. When the bridge is balanced, the two galvanometer terminals are at the same electric potential.
This null method can provide accurate resistance measurements because the balance point is detected at zero galvanometer current.
R₂ / R₁ = Rₓ / R₃
Rₓ = (R₂ × R₃) / R₁
Current through G: non-zero (bridge unbalanced).
Wheatstone Bridge: Step-by-Step
Adjust the resistance until the galvanometer shows zero current.
Example values: R₁ = 4 Ω, R₃ = 6 Ω and Rₓ = 8 Ω. The theoretical balance value is R₂ = (R₁ × Rₓ) / R₃ = 5.33 Ω. Move the slider towards this value.
Teaching model: the needle and current indication are qualitative, not a numerical circuit simulation. At balance, the potential difference between B and D is zero, so no current flows through the galvanometer.
Wheatstone Bridge Formula
Using the resistor labels in the interactive diagram, the balance condition is:
Rearranging to calculate the unknown resistance:
Here, R₁, R₂ and R₃ are known resistances, and Rₓ is the unknown resistance. R₂ is adjusted until the galvanometer reads zero. Always use compatible units for all resistances.
| Symbol | Meaning in the diagram |
|---|---|
| R₁ | Known resistor in the upper-left arm (A–B) |
| R₂ | Known adjustable resistor in the upper-right arm (A–D) |
| R₃ | Known resistor in the lower-left arm (B–C) |
| Rₓ | Unknown resistor in the lower-right arm (D–C) |
Derivation of the Wheatstone Bridge Formula
At balance, no current flows through the galvanometer. Let I₁ be the current in the left branch and I₂ be the current in the right branch. Because the galvanometer terminals B and D are at the same potential, the voltage drops from A to B and from A to D are equal.
Step 1: Apply Ohm’s law to the upper arms.
Therefore, R₂/R₁ = I₁/I₂
Step 2: Apply Ohm’s law to the lower arms.
Therefore, Rₓ/R₃ = I₁/I₂
Step 3: Equate the two ratios. Both ratios equal I₁/I₂, so:
Rearranging gives the unknown resistance:
This derivation applies to the resistor labels shown in the circuit diagram. If a textbook uses different labels or places the unknown resistor in another arm, write the balance ratio from that diagram before rearranging it.
Solved Numerical Examples
Example 1: Find the unknown resistance
Question: A balanced bridge has R₁ = 4 Ω, R₂ = 6 Ω and R₃ = 8 Ω. Find Rₓ.
Step 1 — Formula: Rₓ = (R₂ × R₃) / R₁
Step 2 — Substitute: Rₓ = (6 × 8) / 4 = 48 / 4
Answer: Rₓ = 12 Ω
Example 2: Find a missing known resistance
Question: A balanced bridge has R₁ = 2 Ω, R₃ = 4 Ω and Rₓ = 10 Ω. Find R₂.
Start with R₂/R₁ = Rₓ/R₃. Rearrange to get R₂ = (R₁ × Rₓ) / R₃.
Substitute: R₂ = (2 × 10) / 4 = 20 / 4.
Answer: R₂ = 5 Ω
Example 3: Is the bridge balanced?
Question: R₁ = 3 Ω, R₂ = 6 Ω, R₃ = 4 Ω and Rₓ = 8 Ω. Is the bridge balanced?
Check both ratios: R₂/R₁ = 6/3 = 2, and Rₓ/R₃ = 8/4 = 2.
Answer: Yes. Both ratios are equal, so the bridge is balanced.
Balanced vs Unbalanced Wheatstone Bridge
Unbalanced bridge
The resistance ratios are unequal. Points B and D have different electric potentials, so current flows through the galvanometer and its needle deflects.
Balanced bridge
The resistance ratios are equal. Points B and D have the same electric potential, so the potential difference across the galvanometer is zero and no current flows through it.
Common Mistakes Students Should Avoid
Use the resistor labels from the actual diagram. For this diagram, the balance condition is R₂/R₁ = Rₓ/R₃.
At balance, only the galvanometer-branch current is zero. The two main branch currents can still flow.
Write Ω for ohms or kΩ for kilohms. Keep units compatible throughout a calculation.
The simple formula on this page is used when the bridge is balanced. First check that the two resistance ratios are equal or that the galvanometer reads zero.
Principle and Working
- Connect the circuit: Arrange the four resistances in two branches, connect the supply across A and C, and connect the galvanometer between B and D.
- Observe the galvanometer: If the bridge is unbalanced, a potential difference exists between B and D, so current flows through the galvanometer.
- Adjust a known resistance: Change R₂ to reduce the potential difference and galvanometer current.
- Find the balance point: When the galvanometer reads zero, use the balance formula to calculate Rₓ.
Applications of a Wheatstone Bridge
- Finding unknown resistance: Compare an unknown resistor with known resistances and find the balance point.
- Strain gauges: Detect tiny changes in the shape or length of an object by measuring a change in resistance.
- Temperature sensing: Bridge circuits can detect resistance changes in resistance-based temperature sensors.
- Electronic weighing scales: Strain gauges arranged in a bridge help turn small mechanical changes into electrical signals.
- Physics laboratory work: Students can verify the balance condition and practise accurate resistance measurement.
- Instrumentation: Bridge circuits are used with signal-conditioning electronics to detect small resistance changes.
Note: A basic Wheatstone bridge measures resistance at balance. Sensor systems may use extra electronics to measure and process small changes in resistance.
Limitations of a Wheatstone Bridge
- Lead and contact resistance can introduce errors when measuring very low resistances.
- Temperature changes can alter resistor values and affect the result.
- Very high resistances may be difficult to measure accurately because of leakage currents and detector limitations.
- Good connections, stable resistors and a suitable sensitive detector are needed for reliable measurements.
Practice Questions: Test Your Understanding
Choose an answer for each question, then select Check Answers to see the correct answers and short explanations.
Frequently Asked Questions
What happens when a Wheatstone bridge is balanced?
The two galvanometer terminals have equal potential, so no current flows through the galvanometer branch.
Can an ammeter replace the galvanometer?
An ammeter is designed to measure current and is not usually the preferred null detector for a classic Wheatstone bridge. A sensitive galvanometer or suitable electronic detector is more appropriate.
What is the formula for unknown resistance?
For the resistor labels in this page's diagram, Rₓ = (R₂ × R₃) / R₁ when the bridge is balanced.
What are the main applications of a Wheatstone bridge?
It is used to measure unknown resistance and in sensor circuits such as strain gauges and resistance-based temperature measurement systems.
Why is the Wheatstone bridge accurate?
At balance, the result depends on resistance ratios rather than a direct current reading through the detector. Accuracy still depends on resistor tolerances, temperature stability and good connections.