Resistance Measurements and the Wheatstone Bridge

Resistance Measurements and the Wheatstone Bridge

There is a variety of so-called ohmmeters that purport to measure resistance. What the most common ohmmeters actually do is to apply a voltage to a resistance, measure the current, and calculate the resistance using Ohm’s law. Their readout is this calculated resistance. Two configurations for ohmmeters using standard voltmeters and ammeters are shown in this figure. Such configurations are limited in accuracy, because the meters alter both the voltage applied to the resistor and the current that flows through it.

Two methods for measuring resistance with standard meters. (a) Assuming a known voltage for the source, an ammeter measures current, and resistance is calculated as \(R=\cfrac{V}{I}\). (b) Since the terminal voltage \(V\) varies with current, it is better to measure it. \(V\) is most accurately known when \(I\) is small, but \(I\) itself is most accurately known when it is large.

The Wheatstone bridge is a null measurement device for calculating resistance by balancing potential drops in a circuit. (See this figure.) The device is called a bridge because the galvanometer forms a bridge between two branches. A variety of bridge devices are used to make null measurements in circuits.

Resistors \({R}_{1}\) and \({R}_{2}\) are precisely known, while the arrow through \({R}_{3}\) indicates that it is a variable resistance. The value of \({R}_{3}\) can be precisely read. With the unknown resistance \({R}_{x}\) in the circuit, \({R}_{3}\) is adjusted until the galvanometer reads zero. The potential difference between points b and d is then zero, meaning that b and d are at the same potential. With no current running through the galvanometer, it has no effect on the rest of the circuit. So the branches abc and adc are in parallel, and each branch has the full voltage of the source. That is, the \(\text{IR}\) drops along abc and adc are the same. Since b and d are at the same potential, the \(\text{IR}\) drop along ad must equal the \(\text{IR}\) drop along ab. Thus,

\({I}_{1}{R}_{1}={I}_{2}{R}_{3}.\)

Again, since b and d are at the same potential, the \(\text{IR}\) drop along dc must equal the \(\text{IR}\) drop along bc. Thus,

\({I}_{1}{R}_{2}={I}_{2}{R}_{\text{x}}.\)

Taking the ratio of these last two expressions gives

\(\cfrac{{I}_{1}{R}_{1}}{{I}_{1}{R}_{2}}=\cfrac{{I}_{2}{R}_{3}}{{I}_{2}{R}_{x}}.\)

Canceling the currents and solving for R_x yields

\({R}_{\text{x}}={R}_{3}\cfrac{{R}_{2}}{{R}_{1}}.\)

The Wheatstone bridge is used to calculate unknown resistances. The variable resistance \({R}_{3}\) is adjusted until the galvanometer reads zero with the switch closed. This simplifies the circuit, allowing \({R}_{x}\) to be calculated based on the \(\text{IR}\) drops as discussed in the text.

This equation is used to calculate the unknown resistance when current through the galvanometer is zero. This method can be very accurate (often to four significant digits), but it is limited by two factors. First, it is not possible to get the current through the galvanometer to be exactly zero. Second, there are always uncertainties in \({R}_{1}\), \({R}_{2}\), and \({R}_{3}\), which contribute to the uncertainty in \({R}_{x}\).