Wheatstone bridge

The Wheatstone bridge is a four-resistor circuit arranged in a diamond shape that allows an unknown resistance to be measured with very great precision, by comparing it with known resistances. Its strength lies in a subtle idea: instead of directly measuring a current or a voltage (an operation always affected by instrumental error), one looks for the condition in which a current vanishes — a much sharper and more reliable comparison.

Balance principle

The four resistors form a diamond with vertices A (pole +), C, D and B (pole −). Between C and D a galvanometer $G$ is inserted, an extremely sensitive instrument that detects very small currents. The bridge is in balance when the current in the galvanometer is zero.

Principle — Balance of the Wheatstone bridge

The bridge is balanced when $I_G = 0$. The balance condition is: $\ev{\frac{R_1}{R_2} = \frac{R_3}{R_4}}$

Wheatstone bridge: four resistors in a diamond with a galvanometer $G$ on the central branch and an external source. At balance ($I_G = 0$) $R_1/R_2 = R_3/R_4$ holds.

Derivation of the condition

When $I_G = 0$ no current flows in the central branch: C and D are then at the same potential. The circuit behaves as two independent parallel branches, each carrying a single current:

$i_1 = \frac{\mathcal{E}}{R_1 + R_3} \qquad i_2 = \frac{\mathcal{E}}{R_2 + R_4}$

Since $V_C = V_D$, the potential drops across the two upper resistors $R_1$ and $R_2$ must be equal:

$R_1\,i_1 = R_2\,i_2 \;\Rightarrow\; R_1 \cdot \frac{\mathcal{E}}{R_1+R_3} = R_2 \cdot \frac{\mathcal{E}}{R_2+R_4} \;\Rightarrow\; \frac{R_1}{R_2} = \frac{R_3}{R_4}$

How it is used

In practice $R_1$ and $R_2$ (fixed) are known, and a variable resistor $R_3$, adjustable with precision, is available. $R_3$ is varied until the galvanometer reads zero: once balance is reached, the unknown resistance $R_4$ is obtained from the bridge condition,

$\ev{R_4 = R_3\,\frac{R_2}{R_1}}$

The galvanometer as a null detector

The galvanometer detects currents of the order of micro-amperes: it is the most sensitive instrument in the circuit. It is not used to directly measure a value, but only as a null detector — it serves to establish with great precision when the current vanishes, not how much it is. This is precisely what makes the method so accurate.

Links

Topics: Electric circuits Concepts: Resistors in series and parallel Skills: Network simplification

Related exercises: Worked exercise — mixed series-parallel network · Problem — true or false on series and parallel · Problem — series-parallel equivalent resistance