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.

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

emfxemfs=IRxIRs=RxRs.emfxemfs=IRxIRs=RxRs. size 12{ { {"emf" rSub { size 8{x} } } over {"emf" rSub { size 8{s} } } } = { { ital "IR" rSub { size 8{x} } } over { ital "IR" rSub { size 8{s} } } } = { {R rSub { size 8{x} } } over {R rSub { size 8{s} } } } } {}

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

emfx=emfsRxRs.emfx=emfsRxRs. size 12{"emf" rSub { size 8{x} } ="emf" rSub { size 8{s} } { {R rSub { size 8{x} } } over {R rSub { size 8{s} } } } } {}

(2)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} {}.