Wheatstone Bridge

Worked Example: Analyzing a Wheatstone Bridge

Let's analyze a specific bridge configuration. Consider the following values:

• Top-left resistance, R₁ = 1 kΩ
• Bottom-left resistance, R₂ = 4 kΩ
• Top-right resistance, R₃ = 2 kΩ
• Bottom-right resistance, R₄ = 8 kΩ
• Source voltage, V = 10 V

We want to find the midpoint voltages and whether the bridge is balanced.

1. Calculate the voltage at the left midpoint (V_left):The left side of the bridge acts as a voltage divider with R₁ and R₂.
   V_left = V · (R₂ / (R₁ + R₂))
   V_left = 10 · (4 / (1 + 4)) = 10 · (4 / 5) = 8 V
2. Calculate the voltage at the right midpoint (V_right):The right side is a voltage divider with R₃ and R₄.
   V_right = V · (R₄ / (R₃ + R₄))
   V_right = 10 · (8 / (2 + 8)) = 10 · (8 / 10) = 8 V
3. Determine the meter voltage (V_meter):
   V_meter = V_left - V_right
   V_meter = 8 V - 8 V = 0 V
4. Check the balance condition:The bridge is balanced if the resistance ratios are equal: R₁ · R₄ = R₂ · R₃.
   1 kΩ · 8 kΩ = 4 kΩ · 2 kΩ
   8 = 8. The bridge is balanced, matching the 0 V meter reading.

Interactive Experiment

Explore this concept interactively in the Wheatstone Bridge module.

1. Set Source voltage V to 12.
2. Set Top-left resistance R1 to 4 and Top-right resistance R3 to 1.
3. Set Bottom-left resistance R2 to 2 and Bottom-right resistance R4 to 2.
4. Observe that the bridge is unbalanced (Balanced: False). The left midpoint is at 4V, while the right midpoint is at 8V, causing a meter reading of -4V.
5. Change Top-left resistance R1 to 1 to match R3. Notice how the voltages equalize to 8V on both sides, bringing the meter reading to 0V and balancing the bridge.

Limitations & Assumptions

The model provided here makes several simplifying assumptions:

• Ideal Voltmeter: The meter connecting the two midpoints is assumed to have infinite input impedance, meaning it draws zero current from the bridge. A real galvanometer has finite resistance, which would alter the branch currents when the bridge is unbalanced.
• Ideal Source: The voltage source is assumed to have zero internal resistance.
• No Noise: Sensor noise and thermal fluctuations are ignored.

Additionally, this model defines the "balanced" state purely based on the resistance ratio equality (R₁ · R₄ = R₂ · R₃). Even if the source voltage is set to 0 V (causing all measured voltages and currents to be 0), the bridge is only considered balanced if this resistance ratio holds true.

References

NCERT Physics Textbook: Current Electricity. This excerpt discusses the Wheatstone bridge, its four resistors, an ideal source, and the balance condition. Our module explicitly specializes in an infinite-impedance voltmeter.