Suppose you wish to measure the emf of a battery. Consider what happens if you connect the battery directly to a standard voltmeter as shown in Figure 1. (Once we note the problems with this measurement, we will examine a null measurement that improves accuracy.) As discussed before, the actual quantity measured is the terminal voltage V, which is related to the emf of the battery by V = emf − Ir, where I is the current that flows and r is the internal resistance of the battery. The emf could be accurately calculated if r were very accurately known, but it is usually not. If the current I could be made zero, then V=emf, and so emf could be directly measured. However, standard voltmeters need a current to operate; thus, another technique is needed.

Figure 1. An analog voltmeter attached to a battery draws a small but nonzero current and measures a terminal voltage that differs from the emf of the battery. (Note that the script capital E symbolizes electromotive force, or emf.) Since the internal resistance of the battery is not known precisely, it is not possible to calculate the emf precisely.

A potentiometer is a null measurement device for measuring potentials (voltages). (See Figure 2.) A voltage source is connected to a resistor R, say, a long wire, and passes a constant current through it. There is a steady drop in potential (an IR drop) along the wire, so that a variable potential can be obtained by making contact at varying locations along the wire. Figure 2(b) shows an unknown emf_x (represented by script E_x in the figure) connected in series with a galvanometer. Note that emf_x opposes the other voltage source. The location of the contact point (see the arrow on the drawing) is adjusted until the galvanometer reads zero. When the galvanometer reads zero, emf_x = IR_x, where R_x is the resistance of the section of wire up to the contact point. Since no current flows through the galvanometer, none flows through the unknown emf, and so emf_x is directly sensed. Now, a very precisely known standard emf_s is substituted for emf_x, and the contact point is adjusted until the galvanometer again reads zero, so that emf_s = IR_s. In both cases, no current passes through the galvanometer, and so the current I through the long wire is the same. Upon taking the ratio [latex]\frac{{\text{emf}}_{x}}{{\text{emf}}_{s}}\\[/latex], I cancels, giving

[latex]\frac{{\text{emf}}_{x}}{{\text{emf}}_{s}}=\frac{{IR}_{x}}{{IR}_{s}}=\frac{{R}_{x}}{{R}_{s}}\\[/latex].

Solving for emf_x gives

[latex]{\text{emf}}_{x}={\text{emf}}_{s}\frac{{R}_{x}}{{R}_{s}}\\[/latex].

Figure 2. The potentiometer, a null measurement device. (a) A voltage source connected to a long wire resistor passes a constant current I through it. (b) An unknown emf (labeled script E_x in the figure) is connected as shown, and the point of contact along R is adjusted until the galvanometer reads zero. The segment of wire has a resistance R_x and script E_x, where I is unaffected by the connection since no current flows through the galvanometer. The unknown emf is thus proportional to the resistance of the wire segment.

Because a long uniform wire is used for R, the ratio of resistances R_x/R_s is the same as the ratio of the lengths of wire that zero the galvanometer for each emf. The three quantities on the right-hand side of the equation are now known or measured, and emf_x can be calculated. The uncertainty in this calculation can be considerably smaller than when using a voltmeter directly, but it is not zero. There is always some uncertainty in the ratio of resistances R_x/R_s and in the standard emf_s. Furthermore, it is not possible to tell when the galvanometer reads exactly zero, which introduces error into both R_x and R_s, and may also affect the current I.

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 Figure 3. Such configurations are limited in accuracy, because the meters alter both the voltage applied to the resistor and the current that flows through it.

Figure 3. 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 [latex]R=\frac{V}{I}\\[/latex] . (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 Figure 4.) 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 R₁ and R₂ are precisely known, while the arrow through R₃ indicates that it is a variable resistance. The value of R₃ can be precisely read. With the unknown resistance R_x in the circuit, R₃ 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 IR drops along abc and adc are the same. Since b and d are at the same potential, the IR drop along ad must equal the IR drop along ab.

Thus,

I₁R₁ = I₂R₃.

Again, since b and d are at the same potential, the IR drop along dc must equal the IR drop along bc. Thus,

I₁R₂ = I₂R_x.

Taking the ratio of these last two expressions gives

[latex]\frac{{I}_{1}{R}_{1}}{{I}_{1}{R}_{2}}=\frac{{I}_{2}{R}_{3}}{{I}_{2}{R}_{x}}\\[/latex].

Canceling the currents and solving for R_x yields

[latex]{R}_{\text{x}}={R}_{3}\frac{{R}_{2}}{{R}_{1}}\\[/latex].

Figure 4. The Wheatstone bridge is used to calculate unknown resistances. The variable resistance R₃ is adjusted until the galvanometer reads zero with the switch closed. This simplifies the circuit, allowing R_x to be calculated based on the IR drops as discussed in the text.

Glossary

null measurements:

methods of measuring current and voltage more accurately by balancing the circuit so that no current flows through the measurement device

potentiometer:

a null measurement device for measuring potentials (voltages)

ohmmeter:

an instrument that applies a voltage to a resistance, measures the current, calculates the resistance using Ohm’s law, and provides a readout of this calculated resistance

bridge device:

a device that forms a bridge between two branches of a circuit; some bridge devices are used to make null measurements in circuits

Wheatstone bridge:

a null measurement device for calculating resistance by balancing potential drops in a circuit