Null Measurements

The problem with the kind of voltage measurements we have described so far is that they are really based on magnetism. It turns out that we can only generate magnetic forces if there is at least some current flow. Consequently, for a voltmeter to function, it must allow some current to flow through it instead of flowing through the intended path in the circuit you are measuring. To minimize this unintentional rerouting of current, one tries to make the internal resistance of a voltmeter as large as possible.

We can use a galvanometer (which is always based on magnetism and hence current flow) to push this idea to the extreme with a special configuration that involves a variable resistor, called a potentiometer. The basic insight behind this setup is that there is one particular scenario in which a voltmeter gives a perfectly accurate reading without drawing any current whatsoever: that is when the reading is precisely zero volts. A vanishing voltage means vanishing current by Ohm's Law. Read this text to learn how we can use a technique called a null measurement to measure an unknown emf (voltage).

There is another measurement technique based on drawing no current at all and, hence, not altering the circuit at all. These are called null measurements and are the topic of Null Measurements. Digital meters that employ solid-state electronics and null measurements can attain accuracies of one part in 10^{6}.

Standard measurements of voltage and current alter the circuit being measured, introducing uncertainties in the measurements. Voltmeters draw some extra current, whereas ammeters reduce current flow. Null measurements balance voltages so that there is no current flowing through the measuring device and, therefore, no alteration of the circuit being measured.

Null measurements are generally more accurate but are also more complex than the use of standard voltmeters and ammeters, and they still have limits to their precision. In this module, we shall consider a few specific types of null measurements, because they are common and interesting, and they further illuminate principles of electric circuits.


The Potentiometer

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 21.34. (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.

Schematic of an analog voltmeter attached to a battery

Figure 21.34 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 21.35.) 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 21.35 (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 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 \frac{emf_{x}}{emf_{s}}, I cancels, giving

\frac{emf_{x}}{emf_{s}}=\frac{IR_{x}}{IR_{s}}=\frac{R_{x}}{R_{s}} [Equation 21.71]

Solving for emf_{x} gives

emf_{x}=emf_{s}\frac{R_{x}}{R_{s}} [Equation 21.72]


Schematic of a potentiometer.

Figure 21.35 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}=IR_{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_{x}, and may also affect the current I.


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 Figure 21.36. Such configurations are limited in accuracy, because the meters alter both the voltage applied to the resistor and the current that flows through it.

Schematic of two ammeters.

Figure 21.36 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=\frac{V}{1}. (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 21.37.) 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 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,

I_{1}R_{1}=I_{1}R_{3} [Equation 21.73]

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

I_{1}R_{2}=I_{2}R_{x} [Equation 21.74]

Taking the ratio of these last two expressions gives

\frac{I_{1}R_{1}}{I_{1}R_{2}}=\frac{I_{2}R_{3}}{I_{2}R_{x}} [Equation 21.75]

Canceling the currents and solving for R_{x} yields

R_{x}=R_{3}\frac{R_{2}}{R_{1}} [Equation 21.76]

Schematic of the Wheatstone bridge

Figure 21.37 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 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_{3}.



Rice University, https://openstax.org/books/college-physics/pages/21-5-null-measurements
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Last modified: Tuesday, August 31, 2021, 1:01 PM