The Potentiometer

Suppose you want to determine the emf of a battery as accurately as possible. A simple approach is to connect a voltmeter across the battery, as shown in Figure 31.1. However, as discussed in the previous chapter, the voltmeter measures the terminal voltage, not necessarily the emf.

The measured voltage is related to the battery's emf by

where I is the current drawn from the battery and r is its internal resistance.

If the battery's internal resistance were known perfectly, the emf could be calculated from this equation. In practice, however, the internal resistance is often uncertain and may even change with temperature, age, or battery condition. The most accurate measurement would therefore occur if the current were zero, since then

Unfortunately, ordinary voltmeters require a small current to operate, making a direct measurement impossible. A different approach is needed.

A potentiometer is a null measurement device designed specifically to measure voltage without drawing current from the source being measured. The basic idea is illustrated in Figure 31.2.

A separate voltage source drives a constant current through a long, uniform resistor (often a wire). Because the current remains constant, the voltage decreases uniformly along the length of the wire. By moving a sliding contact, any desired voltage can be selected.

An unknown emf, labeled [latex]{\text{emf}}_x[/latex], is connected in opposition to the voltage developed along the wire through a sensitive galvanometer. The contact point is adjusted until the galvanometer reads zero.

When the galvanometer indicates zero current, no current flows through the unknown battery. Therefore, its terminal voltage equals its emf, giving

where [latex]R_x[/latex] is the resistance of the wire segment between the starting point and the balance point.

Next, the unknown battery is replaced with a standard reference cell having a precisely known emf, [latex]{\text{emf}}_s[/latex]. The contact point is adjusted again until the galvanometer returns to zero, giving

Because the same current flows through the wire during both measurements, taking the ratio eliminates the unknown current:

Solving for the unknown emf gives

Because the resistor is made from a uniform wire, its resistance is directly proportional to its length. Consequently, the resistance ratio [latex]R_x/R_s[/latex] can often be determined simply by measuring the corresponding wire lengths.

This method is considerably more accurate than measuring the voltage directly with a voltmeter because the unknown battery supplies essentially no current during the measurement. Nevertheless, the measurement is not perfect. Small uncertainties remain due to the precision of the reference voltage, the measurement of the wire lengths (or resistances), and the difficulty of determining the exact point where the galvanometer reads zero.

Modern digital instruments often achieve similar results electronically using very high input resistance or automatic null-balancing circuits. These techniques are widely used in laboratory instruments and in medical equipment that measures extremely small electrical signals, such as electrocardiographs (ECGs), electroencephalographs (EEGs), and precision biomedical sensors.

Resistance Measurements and the Wheatstone Bridge

Many instruments sold as ohmmeters appear to measure resistance directly. In reality, most simply apply a known voltage to the unknown resistor, measure the resulting current, and then calculate the resistance using Ohm's law.

Two common methods are illustrated in Figure 31.3. In one configuration, the source voltage is assumed to be known and the current is measured. In the other, both the voltage and current are measured to improve accuracy. Although these methods work well for many applications, they are limited because the voltmeter and ammeter themselves slightly change the circuit being measured.

A much more accurate method is the Wheatstone bridge, a classic example of a null measurement device. Rather than measuring current and voltage directly, the Wheatstone bridge compares an unknown resistance with known resistors and adjusts the circuit until no current flows through a sensitive galvanometer.

The bridge, shown in Figure 31.4, contains three known resistors and one unknown resistance, [latex]R_x[/latex]. The resistor [latex]R_3[/latex] is adjustable, allowing the circuit to be balanced.

When the galvanometer reads zero, no current flows through the bridge branch. At this balance point, points b and d are at the same electric potential. Because no current passes through the galvanometer, it does not affect the rest of the circuit, greatly improving measurement accuracy.

Under these balanced conditions, the voltage drops along corresponding sections of the two branches are equal. Therefore,

Similarly,

Dividing one equation by the other eliminates the unknown currents:

After canceling the currents and solving for the unknown resistance, we obtain the Wheatstone bridge equation:

One advantage of the Wheatstone bridge is that the measuring instrument does not draw current when the bridge is balanced. This eliminates one of the largest sources of measurement error found in conventional voltmeters and ammeters.

For this reason, Wheatstone bridges have been widely used in scientific laboratories and engineering for more than a century. Modern versions remain important in healthcare and biomedical engineering because many sensors—such as strain gauges, pressure transducers, and some temperature sensors—change their resistance in response to physical conditions. Wheatstone bridge circuits can detect these very small resistance changes with extremely high precision, making them useful in devices ranging from blood pressure monitors to electronic laboratory instruments.

Although highly accurate, the Wheatstone bridge is not perfect. Small uncertainties remain because the bridge cannot be balanced with absolute precision, and the known resistors themselves have manufacturing tolerances. Nevertheless, null measurement techniques such as the Wheatstone bridge routinely provide significantly greater accuracy than direct resistance measurements using ordinary meters.