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 VV size 12{V} {}, which is related to the emf of the battery by V=emf−IrV=emf−Ir size 12{V="emf" - ital "Ir"} {}, where II size 12{I} {} is the current that flows and rr size 12{r} {} is the internal resistance of the battery.

The emf could be accurately calculated if rr size 12{r} {} were very accurately known, but it is usually not. If the current II size 12{I} {} could be made zero, then V=emfV=emf size 12{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,R, say, a long wire, and passes a constant current through it. There is a steady drop in potential (an IRIR size 12{ ital "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 emfxemfx size 12{"emf" rSub { size 8{x} } } {} (represented by script ExEx size 12{"emf" rSub { size 8{x} } } {} in the figure) connected in series with a galvanometer. Note that emfxemfx size 12{"emf" rSub { size 8{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, emfx=IRxemfx=IRx size 12{"emf" rSub { size 8{x} } = ital "IR" rSub { size 8{x} } } {}, where RxRx size 12{R rSub { size 8{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 emfxemfx size 12{"emf" rSub { size 8{x} } } {} is directly sensed.

Now, a very precisely known standard emfsemfs size 12{"emf" rSub { size 8{s} } } {} is substituted for emfxemfx size 12{"emf" rSub { size 8{x} } } {}, and the contact point is adjusted until the galvanometer again reads zero, so that emfs=IRsemfs=IRs size 12{"emf" rSub { size 8{s} } = ital "IR" rSub { size 8{s} } } {}. In both cases, no current passes through the galvanometer, and so the current II size 12{I} {} through the long wire is the same. Upon taking the ratio emfxemfs emfxemfs size 12{ { {"emf" rSub { size 8{x} } } over {"emf" rSub { size 8{s} } } } } {}, II size 12{I} {} cancels, giving

Solving for emfxemfx size 12{"emf" rSub { size 8{x} } } {} gives

Figure 2: The potentiometer, a null measurement device. (a) A voltage source connected to a long wire resistor passes a constant current II size 12{I} {} through it. (b) An unknown emf (labeled script ExEx in the figure) is connected as shown, and the point of contact along RR size 12{R} {} is adjusted until the galvanometer reads zero. The segment of wire has a resistance RxRx size 12{R rSub { size 8{x} } } {} and script Ex=IRxEx=IRx size 12{E rSub { size 8{x} } = ital "IR" rSub { size 8{x} } } {}, where II size 12{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 RR size 12{R} {}, the ratio of resistances Rx/RsRx/Rs size 12{R rSub { size 8{x} } /R rSub { size 8{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 emfxemfx size 12{"emf" rSub { size 8{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 Rx/RsRx/Rs size 12{R rSub { size 8{x} } /R rSub { size 8{s} } } {} and in the standard emfsemfs size 12{"emf" rSub { size 8{s} } } {}. Furthermore, it is not possible to tell when the galvanometer reads exactly zero, which introduces error into both RxRx size 12{R rSub { size 8{x} } } {} and RsRs size 12{R rSub { size 8{s} } } {}, and may also affect the current II size 12{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 R=VIR=VI size 12{R= { {V} over {I} } } {}. (b) Since the terminal voltage VV size 12{V} {} varies with current, it is better to measure it. VV size 12{V} {} is most accurately known when II size 12{I} {} is small, but II size 12{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 R1R1 size 12{R rSub { size 8{1} } } {} and R2R2 size 12{R rSub { size 8{2} } } {} are precisely known, while the arrow through R3R3 size 12{R rSub { size 8{3} } } {} indicates that it is a variable resistance. The value of R3R3 size 12{R rSub { size 8{3} } } {} can be precisely read. With the unknown resistance RxRx size 12{R rSub { size 8{x} } } {} in the circuit, R3R3 size 12{R rSub { size 8{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 IRIR size 12{ ital "IR"} {} drops along abc and adc are the same. Since b and d are at the same potential, the IRIR size 12{ ital "IR"} {} drop along ad must equal the IRIR size 12{ ital "IR"} {} drop along ab. Thus,

Again, since b and d are at the same potential, the IRIR size 12{ ital "IR"} {} drop along dc must equal the IRIR size 12{ ital "IR"} {} drop along bc. Thus,

Taking the ratio of these last two expressions gives

Canceling the currents and solving for R_x yields

Figure 4: The Wheatstone bridge is used to calculate unknown resistances. The variable resistance R3R3 size 12{R rSub { size 8{3} } } {} is adjusted until the galvanometer reads zero with the switch closed. This simplifies the circuit, allowing RxRx size 12{R rSub { size 8{x} } } {} to be calculated based on the IRIR size 12{ ital "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 R1R1 size 12{R rSub { size 8{1} } } {}, R2R2 size 12{R rSub { size 8{2} } } {}, and R3R3 size 12{R rSub { size 8{3} } } {}, which contribute to the uncertainty in RxRx size 12{R rSub { size 8{x} } } {}.