{"id":1194,"date":"2017-10-27T16:31:38","date_gmt":"2017-10-27T16:31:38","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/null-measurements\/"},"modified":"2017-11-08T03:26:32","modified_gmt":"2017-11-08T03:26:32","slug":"null-measurements","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/null-measurements\/","title":{"raw":"Null Measurements","rendered":"Null Measurements"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain why a null measurement device is more accurate than a standard voltmeter or ammeter.<\/li>\n<li>Demonstrate how a Wheatstone bridge can be used to accurately calculate the resistance in a circuit.<\/li>\n<\/ul>\n<\/div>\n<p>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. <span data-type=\"term\" id=\"import-auto-id3163834\">Null measurements<\/span> balance voltages so that there is no current flowing through the measuring device and, therefore, no alteration of the circuit being measured.<\/p>\n<p id=\"import-auto-id3258243\">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.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1818552\">\n<h1 data-type=\"title\">The Potentiometer<\/h1>\n<p id=\"import-auto-id1594124\">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 <a href=\"#import-auto-id2691925\" class=\"autogenerated-content\">(Figure)<\/a>. (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 [latex]V[\/latex], which is related to the emf of the battery by [latex]V=\\text{emf}-\\text{Ir}[\/latex], where [latex]I[\/latex] is the current that flows and [latex]r[\/latex] is the internal resistance of the battery.<\/p>\n<p>The emf could be accurately calculated if [latex]r[\/latex] were very accurately known, but it is usually not. If the current [latex]I[\/latex] could be made zero, then [latex]V=\\text{emf}[\/latex], and so emf could be directly measured. However, standard voltmeters need a current to operate; thus, another technique is needed.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2691925\">\n<div class=\"bc-figcaption figcaption\">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.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1599757\" data-alt=\"The diagram shows equivalence between two circuits. The first circuit has a cell of e m f script E and an internal resistance r connected across a voltmeter. The equivalent circuit on the right shows the same cell of e m f script E and an internal resistance r connected across a series combination of a galvanometer with an internal resistance r sub G and high resistance R. The currents in the two circuits are shown to be equal.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_22_05_01.jpg\" data-media-type=\"image\/jpg\" alt=\"The diagram shows equivalence between two circuits. The first circuit has a cell of e m f script E and an internal resistance r connected across a voltmeter. The equivalent circuit on the right shows the same cell of e m f script E and an internal resistance r connected across a series combination of a galvanometer with an internal resistance r sub G and high resistance R. The currents in the two circuits are shown to be equal.\" width=\"275\"><\/span><\/p><\/div>\n<p id=\"import-auto-id506038\">A <span data-type=\"term\">potentiometer<\/span> is a null measurement device for measuring potentials (voltages). (See <a href=\"#import-auto-id3170075\" class=\"autogenerated-content\">(Figure)<\/a>.) A voltage source is connected to a resistor [latex]\\mathrm{R,}[\/latex] say, a long wire, and passes a constant current through it. There is a steady drop in potential (an [latex]\\text{IR}[\/latex] drop) along the wire, so that a variable potential can be obtained by making contact at varying locations along the wire.<\/p>\n<p id=\"import-auto-id3446524\"><a href=\"#import-auto-id3170075\" class=\"autogenerated-content\">(Figure)<\/a>(b) shows an unknown [latex]{\\text{emf}}_{x}[\/latex] (represented by script [latex]{E}_{x}[\/latex]  in the figure) connected in series with a galvanometer. Note that [latex]{\\text{emf}}_{x}[\/latex] 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, [latex]{\\text{emf}}_{x}={\\text{IR}}_{x}[\/latex], where [latex]{R}_{x}[\/latex] 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 [latex]{\\text{emf}}_{x}[\/latex] is directly sensed.<\/p>\n<p id=\"import-auto-id1526247\">Now, a very precisely known standard [latex]{\\text{emf}}_{s}[\/latex] is substituted for [latex]{\\text{emf}}_{x}[\/latex], and the contact point is adjusted until the galvanometer again reads zero, so that [latex]{\\text{emf}}_{s}={\\text{IR}}_{s}[\/latex]. In both cases, no current passes through the galvanometer, and so the current [latex]I[\/latex] through the long wire is the same. Upon taking the ratio [latex]\\frac{{\\text{emf}}_{x}}{{\\text{emf}}_{s}}[\/latex], [latex]I[\/latex] cancels, giving<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\frac{{\\text{emf}}_{x}}{{\\text{emf}}_{s}}=\\frac{{\\text{IR}}_{x}}{{\\text{IR}}_{s}}=\\frac{{R}_{x}}{{R}_{s}}.[\/latex]<\/div>\n<p>Solving for [latex]{\\text{emf}}_{x}[\/latex] gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\text{emf}}_{x}={\\text{emf}}_{s}\\frac{{R}_{x}}{{R}_{s}}.[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3170075\">\n<div class=\"bc-figcaption figcaption\">The potentiometer, a null measurement device. (a) A voltage source connected to a long wire resistor passes a constant current [latex]I[\/latex] through it. (b) An unknown emf (labeled script [latex]{E}_{\\text{x}}[\/latex] in the figure) is connected as shown, and the point of contact along [latex]R[\/latex] is adjusted until the galvanometer reads zero. The segment of wire has a resistance [latex]{R}_{\\text{x}}[\/latex] and script [latex]{E}_{\\text{x}}={\\text{IR}}_{\\text{x}}[\/latex], where [latex]I[\/latex] 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.<\/div>\n<p><span data-type=\"media\" data-alt=\"Two circuits are shown. The first circuit has a cell of e m f script E and internal resistance r connected in series to a resistor R. The second diagram shows the same circuit with the addition of a galvanometer and unknown voltage source connected with a variable contact that can be adjusted up and down the length of the resistor R.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_22_05_02.jpg\" data-media-type=\"image\/jpg\" alt=\"Two circuits are shown. The first circuit has a cell of e m f script E and internal resistance r connected in series to a resistor R. The second diagram shows the same circuit with the addition of a galvanometer and unknown voltage source connected with a variable contact that can be adjusted up and down the length of the resistor R.\" width=\"225\"><\/span><\/p><\/div>\n<p>Because a long uniform wire is used for [latex]R[\/latex], the ratio of resistances [latex]{R}_{\\text{x}}\/{R}_{\\text{s}}[\/latex] 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 [latex]{\\text{emf}}_{\\text{x}}[\/latex] 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 [latex]{R}_{\\text{x}}\/{R}_{\\text{s}}[\/latex] and in the standard [latex]{\\text{emf}}_{s}[\/latex]. Furthermore, it is not possible to tell when the galvanometer reads exactly zero, which introduces error into both [latex]{R}_{\\text{x}}[\/latex] and [latex]{R}_{\\text{s}}[\/latex], and may also affect the current [latex]I[\/latex].<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3013426\">\n<h1 data-type=\"title\">Resistance Measurements and the Wheatstone Bridge<\/h1>\n<p id=\"import-auto-id1221104\">There is a variety of so-called <span data-type=\"term\">ohmmeters<\/span> 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\u2019s law. Their readout is this calculated resistance. Two configurations for ohmmeters using standard voltmeters and ammeters are shown in <a href=\"#import-auto-id3027669\" class=\"autogenerated-content\">(Figure)<\/a>. Such configurations are limited in accuracy, because the meters alter both the voltage applied to the resistor and the current that flows through it.<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">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 [latex]V[\/latex] varies with current, it is better to measure it. [latex]V[\/latex] is most accurately known when [latex]I[\/latex] is small, but [latex]I[\/latex] itself is most accurately known when it is large.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2405734\" data-alt=\"The diagram shows two circuits. The first one has a cell of e m f script E and internal resistance r connected in series to an ammeter A and a resistor R. The second circuit is the same as the first, but in addition there is a voltmeter connected across the voltage source E.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_22_05_03.jpg\" data-media-type=\"image\/jpg\" alt=\"The diagram shows two circuits. The first one has a cell of e m f script E and internal resistance r connected in series to an ammeter A and a resistor R. The second circuit is the same as the first, but in addition there is a voltmeter connected across the voltage source E.\" width=\"275\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3306053\">The <span data-type=\"term\">Wheatstone bridge<\/span> is a null measurement device for calculating resistance by balancing potential drops in a circuit. (See <a href=\"#import-auto-id2446499\" class=\"autogenerated-content\">(Figure)<\/a>.) The device is called a bridge because the galvanometer forms a bridge between two branches. A variety of <span data-type=\"term\">bridge devices<\/span> are used to make null measurements in circuits.<\/p>\n<p id=\"import-auto-id2640034\">Resistors [latex]{R}_{1}[\/latex] and [latex]{R}_{2}[\/latex] are precisely known, while the arrow through [latex]{R}_{3}[\/latex] indicates that it is a variable resistance. The value of [latex]{R}_{3}[\/latex] can be precisely read. With the unknown resistance [latex]{R}_{x}[\/latex] in the circuit, [latex]{R}_{3}[\/latex] 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 [latex]\\text{IR}[\/latex] drops along abc and adc are the same. Since b and d are at the same potential, the [latex]\\text{IR}[\/latex] drop along ad must equal the [latex]\\text{IR}[\/latex] drop along ab. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{I}_{1}{R}_{1}={I}_{2}{R}_{3}.[\/latex]<\/div>\n<p id=\"import-auto-id1019862\">Again, since b and d are at the same potential, the [latex]\\text{IR}[\/latex] drop along dc must equal the [latex]\\text{IR}[\/latex] drop along bc. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-892\">[latex]{I}_{1}{R}_{2}={I}_{2}{R}_{\\text{x}}.[\/latex]<\/div>\n<p>Taking the ratio of these last two expressions gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-737\">[latex]\\frac{{I}_{1}{R}_{1}}{{I}_{1}{R}_{2}}=\\frac{{I}_{2}{R}_{3}}{{I}_{2}{R}_{x}}.[\/latex]<\/div>\n<p id=\"import-auto-id3011011\">Canceling the currents and solving for R<sub>x<\/sub> yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-163\">[latex]{R}_{\\text{x}}={R}_{3}\\frac{{R}_{2}}{{R}_{1}}.[\/latex]<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2446499\">\n<div class=\"bc-figcaption figcaption\">The Wheatstone bridge is used to calculate unknown resistances. The variable resistance [latex]{R}_{3}[\/latex] is adjusted until the galvanometer reads zero with the switch closed. This simplifies the circuit, allowing [latex]{R}_{x}[\/latex] to be calculated based on the [latex]\\text{IR}[\/latex] drops as discussed in the text.<\/div>\n<p><span data-type=\"media\" data-alt=\"This complex circuit diagram shows a galvanometer connected in the center arm of a Wheatstone bridge arrangement. All the other four arms have a resistor. The bridge is connected to a cell of e m f script E and internal resistance r.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_22_05_04.jpg\" data-media-type=\"image\/jpg\" alt=\"This complex circuit diagram shows a galvanometer connected in the center arm of a Wheatstone bridge arrangement. All the other four arms have a resistor. The bridge is connected to a cell of e m f script E and internal resistance r.\" width=\"225\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2402191\">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 [latex]{R}_{1}[\/latex], [latex]{R}_{2}[\/latex], and [latex]{R}_{3}[\/latex], which contribute to the uncertainty in [latex]{R}_{x}[\/latex].<\/p>\n<div data-type=\"exercise\" class=\"exercise\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1389752\">Identify other factors that might limit the accuracy of null measurements. Would the use of a digital device that is more sensitive than a galvanometer improve the accuracy of null measurements?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2442041\" data-print-placement=\"here\">\n<p id=\"import-auto-id2661791\">One factor would be resistance in the wires and connections in a null measurement. These are impossible to make zero, and they can change over time. Another factor would be temperature variations in resistance, which can be reduced but not completely eliminated by choice of material. Digital devices sensitive to smaller currents than analog devices do improve the accuracy of null measurements because they allow you to get the current closer to zero.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1933205\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul>\n<li id=\"import-auto-id2396485\">Null measurement techniques achieve greater accuracy by balancing a circuit so that no current flows through the measuring device.<\/li>\n<li id=\"import-auto-id2971477\">One such device, for determining voltage, is a potentiometer.<\/li>\n<li id=\"import-auto-id1596308\">Another null measurement device, for determining resistance, is the Wheatstone bridge.<\/li>\n<li id=\"import-auto-id2648444\">Other physical quantities can also be measured with null measurement techniques.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id3260722\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1053475\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1352487\">\n<p id=\"import-auto-id3009592\">Why can a null measurement be more accurate than one using standard voltmeters and ammeters? What factors limit the accuracy of null measurements?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3077174\">\n<p id=\"import-auto-id2973094\">If a potentiometer is used to measure cell emfs on the order of a few volts, why is it most accurate for the standard [latex]{\\text{emf}}_{\\text{s}}[\/latex] to be the same order of magnitude and the resistances to be in the range of a few ohms?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1934556\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3104476\">\n<p>What is the [latex]{\\text{emf}}_{\\text{x}}[\/latex] of a cell being measured in a potentiometer, if the standard cell\u2019s emf is 12.0 V and the potentiometer balances for [latex]{R}_{\\text{x}}=5\\text{.}\\text{000}\\phantom{\\rule{0.15em}{0ex}}\\Omega [\/latex] and [latex]{R}_{\\text{s}}=2\\text{.}\\text{500}\\phantom{\\rule{0.15em}{0ex}}\\Omega [\/latex]?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id1860120\">24.0 V<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2589124\">\n<p id=\"import-auto-id1282140\">Calculate the [latex]{\\text{emf}}_{\\text{x}}[\/latex] of a dry cell for which a potentiometer is balanced when [latex]{R}_{\\text{x}}=1\\text{.}\\text{200}\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex], while an alkaline standard cell with an emf of 1.600 V requires [latex]{R}_{\\text{s}}=1\\text{.}\\text{247}\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex] to balance the potentiometer.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1422615\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id2453563\">When an unknown resistance [latex]{R}_{\\text{x}}[\/latex] is placed in a Wheatstone bridge, it is possible to balance the bridge by adjusting [latex]{R}_{3}[\/latex] to be [latex]\\text{2500}\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex]. What is [latex]{R}_{\\text{x}}[\/latex] if [latex]\\frac{{R}_{2}}{{R}_{1}}=0\\text{.}\\text{625}[\/latex]?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id2688500\">[latex]1\\text{.}\\text{56 k}\\Omega [\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2929411\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1427597\">\n<p id=\"import-auto-id3011035\">To what value must you adjust [latex]{R}_{3}[\/latex] to balance a Wheatstone bridge, if the unknown resistance [latex]{R}_{\\text{x}}[\/latex] is [latex]\\text{100}\\phantom{\\rule{0.15em}{0ex}}\\Omega [\/latex], [latex]{R}_{1}[\/latex] is [latex]\\text{50}\\text{.}0\\phantom{\\rule{0.15em}{0ex}}\\Omega [\/latex], and [latex]{R}_{2}[\/latex] is [latex]\\text{175}\\phantom{\\rule{0.15em}{0ex}}\\Omega [\/latex]?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2391704\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1360568\">\n<p id=\"import-auto-id2000805\">(a) What is the unknown [latex]{\\text{emf}}_{\\text{x}}[\/latex] in a potentiometer that balances when [latex]{R}_{\\text{x}}[\/latex] is [latex]\\text{10}\\text{.}0\\phantom{\\rule{0.15em}{0ex}}\\Omega [\/latex], and balances when [latex]{R}_{\\text{s}}[\/latex] is [latex]\\text{15}\\text{.}0\\phantom{\\rule{0.15em}{0ex}}\\Omega [\/latex] for a standard 3.000-V emf? (b) The same [latex]{\\text{emf}}_{\\text{x}}[\/latex] is placed in the same potentiometer, which now balances when [latex]{R}_{\\text{s}}[\/latex] is [latex]\\text{15}\\text{.}0\\phantom{\\rule{0.15em}{0ex}}\\Omega [\/latex] for a standard emf of 3.100 V. At what resistance [latex]{R}_{\\text{x}}[\/latex] will the potentiometer balance?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1994680\">\n<p id=\"import-auto-id1374042\">(a) 2.00 V<\/p>\n<p id=\"import-auto-id1933671\">(b) [latex]9\\text{.}\\text{68}\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3335517\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2678317\">\n<p>Suppose you want to measure resistances in the range from [latex]\\text{10}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex] to [latex]\\text{10}\\text{.}0 k\\Omega [\/latex] using a Wheatstone bridge that has [latex]\\frac{{R}_{2}}{{R}_{1}}=2\\text{.}\\text{000}[\/latex]. Over what range should [latex]{R}_{3}[\/latex] be adjustable?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1352332\">[latex]\\text{Range = 5}\\text{.}\\text{00}\\phantom{\\rule{0.25em}{0ex}}\\Omega \\phantom{\\rule{0.25em}{0ex}}\\text{to}\\phantom{\\rule{0.25em}{0ex}}5\\text{.}\\text{00}\\phantom{\\rule{0.25em}{0ex}}\\text{k}\\Omega [\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id3064821\">\n<dt>null measurements<\/dt>\n<dd id=\"fs-id3033434\">methods of measuring current and voltage more accurately by balancing the circuit so that no current flows through the measurement device<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3077208\">\n<dt>potentiometer<\/dt>\n<dd id=\"fs-id2639302\">a null measurement device for measuring potentials (voltages)<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2447682\">\n<dt>ohmmeter<\/dt>\n<dd id=\"fs-id3103221\">an instrument that applies a voltage to a resistance, measures the current, calculates the resistance using Ohm\u2019s law, and provides a readout of this calculated resistance<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2391549\">\n<dt>bridge device<\/dt>\n<dd>a device that forms a bridge between two branches of a circuit; some bridge devices are used to make null measurements in circuits<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Wheatstone bridge<\/dt>\n<dd id=\"fs-id3285619\">a null measurement device for calculating resistance by balancing potential drops in a circuit<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain why a null measurement device is more accurate than a standard voltmeter or ammeter.<\/li>\n<li>Demonstrate how a Wheatstone bridge can be used to accurately calculate the resistance in a circuit.<\/li>\n<\/ul>\n<\/div>\n<p>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. <span data-type=\"term\" id=\"import-auto-id3163834\">Null measurements<\/span> balance voltages so that there is no current flowing through the measuring device and, therefore, no alteration of the circuit being measured.<\/p>\n<p id=\"import-auto-id3258243\">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.<\/p>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1818552\">\n<h1 data-type=\"title\">The Potentiometer<\/h1>\n<p id=\"import-auto-id1594124\">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 <a href=\"#import-auto-id2691925\" class=\"autogenerated-content\">(Figure)<\/a>. (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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, which is related to the emf of the battery by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-569f10348e902b6282e5b251abf271df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#114;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"101\" style=\"vertical-align: -1px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the current that flows and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> is the internal resistance of the battery.<\/p>\n<p>The emf could be accurately calculated if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c409433a9e2dfcdb83360a974d243f18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"8\" style=\"vertical-align: 0px;\" \/> were very accurately known, but it is usually not. If the current <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> could be made zero, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c90e381dd44a0497a9916ef96d32736_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"68\" style=\"vertical-align: -1px;\" \/>, and so emf could be directly measured. However, standard voltmeters need a current to operate; thus, another technique is needed.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2691925\">\n<div class=\"bc-figcaption figcaption\">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.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1599757\" data-alt=\"The diagram shows equivalence between two circuits. The first circuit has a cell of e m f script E and an internal resistance r connected across a voltmeter. The equivalent circuit on the right shows the same cell of e m f script E and an internal resistance r connected across a series combination of a galvanometer with an internal resistance r sub G and high resistance R. The currents in the two circuits are shown to be equal.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_22_05_01.jpg\" data-media-type=\"image\/jpg\" alt=\"The diagram shows equivalence between two circuits. The first circuit has a cell of e m f script E and an internal resistance r connected across a voltmeter. The equivalent circuit on the right shows the same cell of e m f script E and an internal resistance r connected across a series combination of a galvanometer with an internal resistance r sub G and high resistance R. The currents in the two circuits are shown to be equal.\" width=\"275\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id506038\">A <span data-type=\"term\">potentiometer<\/span> is a null measurement device for measuring potentials (voltages). (See <a href=\"#import-auto-id3170075\" class=\"autogenerated-content\">(Figure)<\/a>.) A voltage source is connected to a resistor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-79e4d432397c0919c65188ca144d0cf8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#82;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"17\" style=\"vertical-align: -4px;\" \/> say, a long wire, and passes a constant current through it. There is a steady drop in potential (an <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b198d7637608c8b42a36583cdda301f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"19\" style=\"vertical-align: -1px;\" \/> drop) along the wire, so that a variable potential can be obtained by making contact at varying locations along the wire.<\/p>\n<p id=\"import-auto-id3446524\"><a href=\"#import-auto-id3170075\" class=\"autogenerated-content\">(Figure)<\/a>(b) shows an unknown <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c60c57950bf31ec5617ff94fb6aca7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"36\" style=\"vertical-align: -3px;\" \/> (represented by script <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-74668dcdea6de641c5df6cbddde0e24c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/>  in the figure) connected in series with a galvanometer. Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c60c57950bf31ec5617ff94fb6aca7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"36\" style=\"vertical-align: -3px;\" \/> 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, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ca2382261e1b926e00acc6f8b6c47ef9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6d145dc4cb226f69d6e9bfe4abf13775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> 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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c60c57950bf31ec5617ff94fb6aca7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"36\" style=\"vertical-align: -3px;\" \/> is directly sensed.<\/p>\n<p id=\"import-auto-id1526247\">Now, a very precisely known standard <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9f8ba90f0fa0d54d9d1da82e86d5940f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"34\" style=\"vertical-align: -3px;\" \/> is substituted for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c60c57950bf31ec5617ff94fb6aca7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"36\" style=\"vertical-align: -3px;\" \/>, and the contact point is adjusted until the galvanometer again reads zero, so that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-34d58d03e1fc36d527a2ad83ce8e0f63_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#115;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -3px;\" \/>. In both cases, no current passes through the galvanometer, and so the current <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> through the long wire is the same. Upon taking the ratio <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7e3974c2b00ea69f19ec8e06e134bbac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;&#125;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"30\" style=\"vertical-align: -8px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> cancels, giving<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-05985c6bb8eab259e99a1c25a5765fbd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;&#125;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#115;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;&#125;&#95;&#123;&#120;&#125;&#125;&#123;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;&#125;&#95;&#123;&#115;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#125;&#123;&#123;&#82;&#125;&#95;&#123;&#115;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"131\" style=\"vertical-align: -8px;\" \/><\/div>\n<p>Solving for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c60c57950bf31ec5617ff94fb6aca7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"36\" style=\"vertical-align: -3px;\" \/> gives<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d0618864ee14ccccdd96a694abef7017_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#120;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#115;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#125;&#123;&#123;&#82;&#125;&#95;&#123;&#115;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"121\" style=\"vertical-align: -8px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id3170075\">\n<div class=\"bc-figcaption figcaption\">The potentiometer, a null measurement device. (a) A voltage source connected to a long wire resistor passes a constant current <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> through it. (b) An unknown emf (labeled script <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95e98b7924a5427cc53957f7ab7860a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> in the figure) is connected as shown, and the point of contact along <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is adjusted until the galvanometer reads zero. The segment of wire has a resistance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44610facafc478b2f9a781c525158d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> and script <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5bf30827af6c760fec3b2de7fbacaeeb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#69;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> 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.<\/div>\n<p><span data-type=\"media\" data-alt=\"Two circuits are shown. The first circuit has a cell of e m f script E and internal resistance r connected in series to a resistor R. The second diagram shows the same circuit with the addition of a galvanometer and unknown voltage source connected with a variable contact that can be adjusted up and down the length of the resistor R.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_22_05_02.jpg\" data-media-type=\"image\/jpg\" alt=\"Two circuits are shown. The first circuit has a cell of e m f script E and internal resistance r connected in series to a resistor R. The second diagram shows the same circuit with the addition of a galvanometer and unknown voltage source connected with a variable contact that can be adjusted up and down the length of the resistor R.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<p>Because a long uniform wire is used for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>, the ratio of resistances <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7f7c88897e74f7e2687a3223b97fe2ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#47;&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -5px;\" \/> 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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7004c71b2a1aa0422b6198cecf71027a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"36\" style=\"vertical-align: -4px;\" \/> 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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7f7c88897e74f7e2687a3223b97fe2ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#47;&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"49\" style=\"vertical-align: -5px;\" \/> and in the standard <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9f8ba90f0fa0d54d9d1da82e86d5940f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"34\" style=\"vertical-align: -3px;\" \/>. Furthermore, it is not possible to tell when the galvanometer reads exactly zero, which introduces error into both <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44610facafc478b2f9a781c525158d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5da4eb29d152d356fa7270280dbf5eb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/>, and may also affect the current <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3013426\">\n<h1 data-type=\"title\">Resistance Measurements and the Wheatstone Bridge<\/h1>\n<p id=\"import-auto-id1221104\">There is a variety of so-called <span data-type=\"term\">ohmmeters<\/span> 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\u2019s law. Their readout is this calculated resistance. Two configurations for ohmmeters using standard voltmeters and ammeters are shown in <a href=\"#import-auto-id3027669\" class=\"autogenerated-content\">(Figure)<\/a>. Such configurations are limited in accuracy, because the meters alter both the voltage applied to the resistor and the current that flows through it.<\/p>\n<div class=\"bc-figure figure\">\n<div class=\"bc-figcaption figcaption\">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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2c9a4032425362b40382512c18c51503_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#86;&#125;&#123;&#73;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"51\" style=\"vertical-align: -6px;\" \/>. (b) Since the terminal voltage <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> varies with current, it is better to measure it. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is most accurately known when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> is small, but <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-18b5e45cb4a1ee02e81b9a980f828db8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\" \/> itself is most accurately known when it is large.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2405734\" data-alt=\"The diagram shows two circuits. The first one has a cell of e m f script E and internal resistance r connected in series to an ammeter A and a resistor R. The second circuit is the same as the first, but in addition there is a voltmeter connected across the voltage source E.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_22_05_03.jpg\" data-media-type=\"image\/jpg\" alt=\"The diagram shows two circuits. The first one has a cell of e m f script E and internal resistance r connected in series to an ammeter A and a resistor R. The second circuit is the same as the first, but in addition there is a voltmeter connected across the voltage source E.\" width=\"275\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3306053\">The <span data-type=\"term\">Wheatstone bridge<\/span> is a null measurement device for calculating resistance by balancing potential drops in a circuit. (See <a href=\"#import-auto-id2446499\" class=\"autogenerated-content\">(Figure)<\/a>.) The device is called a bridge because the galvanometer forms a bridge between two branches. A variety of <span data-type=\"term\">bridge devices<\/span> are used to make null measurements in circuits.<\/p>\n<p id=\"import-auto-id2640034\">Resistors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0d19d200b200b540404daced90cb6826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0915a0a0df1a23a75160d4e10dd1e0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> are precisely known, while the arrow through <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95623682ab8761d5c5c37a562f2a73d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> indicates that it is a variable resistance. The value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95623682ab8761d5c5c37a562f2a73d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> can be precisely read. With the unknown resistance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6d145dc4cb226f69d6e9bfe4abf13775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> in the circuit, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95623682ab8761d5c5c37a562f2a73d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> 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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b198d7637608c8b42a36583cdda301f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"19\" style=\"vertical-align: -1px;\" \/> drops along abc and adc are the same. Since b and d are at the same potential, the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b198d7637608c8b42a36583cdda301f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"19\" style=\"vertical-align: -1px;\" \/> drop along ad must equal the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b198d7637608c8b42a36583cdda301f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"19\" style=\"vertical-align: -1px;\" \/> drop along ab. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-edb13f445af087e6a88f4e5ed0b9ff6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#73;&#125;&#95;&#123;&#49;&#125;&#123;&#82;&#125;&#95;&#123;&#49;&#125;&#61;&#123;&#73;&#125;&#95;&#123;&#50;&#125;&#123;&#82;&#125;&#95;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1019862\">Again, since b and d are at the same potential, the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b198d7637608c8b42a36583cdda301f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"19\" style=\"vertical-align: -1px;\" \/> drop along dc must equal the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b198d7637608c8b42a36583cdda301f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"19\" style=\"vertical-align: -1px;\" \/> drop along bc. Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-892\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-222cbfb5d297f0600980b416e889b5bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#73;&#125;&#95;&#123;&#49;&#125;&#123;&#82;&#125;&#95;&#123;&#50;&#125;&#61;&#123;&#73;&#125;&#95;&#123;&#50;&#125;&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>Taking the ratio of these last two expressions gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-737\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-af5fbc456eeaefbcd1b005cb94cba142_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#73;&#125;&#95;&#123;&#49;&#125;&#123;&#82;&#125;&#95;&#123;&#49;&#125;&#125;&#123;&#123;&#73;&#125;&#95;&#123;&#49;&#125;&#123;&#82;&#125;&#95;&#123;&#50;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#73;&#125;&#95;&#123;&#50;&#125;&#123;&#82;&#125;&#95;&#123;&#51;&#125;&#125;&#123;&#123;&#73;&#125;&#95;&#123;&#50;&#125;&#123;&#82;&#125;&#95;&#123;&#120;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"94\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id3011011\">Canceling the currents and solving for R<sub>x<\/sub> yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-163\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6cfe9c5d4746ed1b372ad6f0f548955d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#61;&#123;&#82;&#125;&#95;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#82;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#123;&#82;&#125;&#95;&#123;&#49;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"91\" style=\"vertical-align: -9px;\" \/><\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2446499\">\n<div class=\"bc-figcaption figcaption\">The Wheatstone bridge is used to calculate unknown resistances. The variable resistance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95623682ab8761d5c5c37a562f2a73d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> is adjusted until the galvanometer reads zero with the switch closed. This simplifies the circuit, allowing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6d145dc4cb226f69d6e9bfe4abf13775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> to be calculated based on the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7b198d7637608c8b42a36583cdda301f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#73;&#82;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"19\" style=\"vertical-align: -1px;\" \/> drops as discussed in the text.<\/div>\n<p><span data-type=\"media\" data-alt=\"This complex circuit diagram shows a galvanometer connected in the center arm of a Wheatstone bridge arrangement. All the other four arms have a resistor. The bridge is connected to a cell of e m f script E and internal resistance r.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_22_05_04.jpg\" data-media-type=\"image\/jpg\" alt=\"This complex circuit diagram shows a galvanometer connected in the center arm of a Wheatstone bridge arrangement. All the other four arms have a resistor. The bridge is connected to a cell of e m f script E and internal resistance r.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2402191\">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 <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0d19d200b200b540404daced90cb6826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0915a0a0df1a23a75160d4e10dd1e0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95623682ab8761d5c5c37a562f2a73d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/>, which contribute to the uncertainty in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6d145dc4cb226f69d6e9bfe4abf13775_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#120;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/>.<\/p>\n<div data-type=\"exercise\" class=\"exercise\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id1389752\">Identify other factors that might limit the accuracy of null measurements. Would the use of a digital device that is more sensitive than a galvanometer improve the accuracy of null measurements?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2442041\" data-print-placement=\"here\">\n<p id=\"import-auto-id2661791\">One factor would be resistance in the wires and connections in a null measurement. These are impossible to make zero, and they can change over time. Another factor would be temperature variations in resistance, which can be reduced but not completely eliminated by choice of material. Digital devices sensitive to smaller currents than analog devices do improve the accuracy of null measurements because they allow you to get the current closer to zero.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1933205\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul>\n<li id=\"import-auto-id2396485\">Null measurement techniques achieve greater accuracy by balancing a circuit so that no current flows through the measuring device.<\/li>\n<li id=\"import-auto-id2971477\">One such device, for determining voltage, is a potentiometer.<\/li>\n<li id=\"import-auto-id1596308\">Another null measurement device, for determining resistance, is the Wheatstone bridge.<\/li>\n<li id=\"import-auto-id2648444\">Other physical quantities can also be measured with null measurement techniques.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id3260722\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1053475\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1352487\">\n<p id=\"import-auto-id3009592\">Why can a null measurement be more accurate than one using standard voltmeters and ammeters? What factors limit the accuracy of null measurements?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3077174\">\n<p id=\"import-auto-id2973094\">If a potentiometer is used to measure cell emfs on the order of a few volts, why is it most accurate for the standard <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-575999db45e564daaf6dd6ba230917e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"33\" style=\"vertical-align: -3px;\" \/> to be the same order of magnitude and the resistances to be in the range of a few ohms?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1934556\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problem Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3104476\">\n<p>What is the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7004c71b2a1aa0422b6198cecf71027a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"36\" style=\"vertical-align: -4px;\" \/> of a cell being measured in a potentiometer, if the standard cell\u2019s emf is 12.0 V and the potentiometer balances for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6470789d707295fe18d6ec7a7dcad079_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#61;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9d9524b138e80df4f486e208d0af3f2c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"98\" style=\"vertical-align: -3px;\" \/>?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id1860120\">24.0 V<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2589124\">\n<p id=\"import-auto-id1282140\">Calculate the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7004c71b2a1aa0422b6198cecf71027a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"36\" style=\"vertical-align: -4px;\" \/> of a dry cell for which a potentiometer is balanced when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0a764ee0b1f924afcc8e1a7d4da93ad0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"102\" style=\"vertical-align: -4px;\" \/>, while an alkaline standard cell with an emf of 1.600 V requires <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8797e74cf107f4de1244269eaed88076_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"100\" style=\"vertical-align: -3px;\" \/> to balance the potentiometer.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1422615\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\">\n<p id=\"import-auto-id2453563\">When an unknown resistance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44610facafc478b2f9a781c525158d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> is placed in a Wheatstone bridge, it is possible to balance the bridge by adjusting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95623682ab8761d5c5c37a562f2a73d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b9495518430cbac93be20325d9005f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"52\" style=\"vertical-align: 0px;\" \/>. What is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44610facafc478b2f9a781c525158d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1c3e16ca66b86c5e2736ddb75867a58b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#82;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#123;&#82;&#125;&#95;&#123;&#49;&#125;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"82\" style=\"vertical-align: -9px;\" \/>?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<p id=\"import-auto-id2688500\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-455e280da0e769696b033af77fe1b082_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#54;&#32;&#107;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2929411\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1427597\">\n<p id=\"import-auto-id3011035\">To what value must you adjust <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95623682ab8761d5c5c37a562f2a73d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> to balance a Wheatstone bridge, if the unknown resistance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44610facafc478b2f9a781c525158d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-38433fd11ac1afed6f94e94816bee76c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0d19d200b200b540404daced90cb6826_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c4810018c699f93a7398de8781f5e100_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0915a0a0df1a23a75160d4e10dd1e0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f33b8aabf850e92920442d956c491cae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#55;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"40\" style=\"vertical-align: -1px;\" \/>?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2391704\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1360568\">\n<p id=\"import-auto-id2000805\">(a) What is the unknown <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7004c71b2a1aa0422b6198cecf71027a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"36\" style=\"vertical-align: -4px;\" \/> in a potentiometer that balances when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44610facafc478b2f9a781c525158d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2dde4556ccafad95d0fed859cf29d0cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: -1px;\" \/>, and balances when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5da4eb29d152d356fa7270280dbf5eb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-45f360f24ef30418da178ce9f2d285c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -1px;\" \/> for a standard 3.000-V emf? (b) The same <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7004c71b2a1aa0422b6198cecf71027a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#101;&#109;&#102;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"36\" style=\"vertical-align: -4px;\" \/> is placed in the same potentiometer, which now balances when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5da4eb29d152d356fa7270280dbf5eb7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"18\" style=\"vertical-align: -3px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-45f360f24ef30418da178ce9f2d285c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#49;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"45\" style=\"vertical-align: -1px;\" \/> for a standard emf of 3.100 V. At what resistance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44610facafc478b2f9a781c525158d92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#120;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"21\" style=\"vertical-align: -4px;\" \/> will the potentiometer balance?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1994680\">\n<p id=\"import-auto-id1374042\">(a) 2.00 V<\/p>\n<p id=\"import-auto-id1933671\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9ed544ae28287ade34080bf65fe6fee1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"48\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3335517\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2678317\">\n<p>Suppose you want to measure resistances in the range from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-21a09b5b5d35acb6100a3e0f8b55ec59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/> to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b024e4c149770950f6d63fb7fb5e97b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#32;&#107;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"52\" style=\"vertical-align: -1px;\" \/> using a Wheatstone bridge that has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-23d3eb75b7a8ffeaa0880e149edc9a2b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#82;&#125;&#95;&#123;&#50;&#125;&#125;&#123;&#123;&#82;&#125;&#95;&#123;&#49;&#125;&#125;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"83\" style=\"vertical-align: -9px;\" \/>. Over what range should <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95623682ab8761d5c5c37a562f2a73d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> be adjustable?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\">\n<div data-type=\"equation\" class=\"equation\" id=\"eip-id1352332\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-68300806449576130601d8033318302b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#97;&#110;&#103;&#101;&#32;&#61;&#32;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#111;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#125;&#92;&#79;&#109;&#101;&#103;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"205\" style=\"vertical-align: -3px;\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id3064821\">\n<dt>null measurements<\/dt>\n<dd id=\"fs-id3033434\">methods of measuring current and voltage more accurately by balancing the circuit so that no current flows through the measurement device<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3077208\">\n<dt>potentiometer<\/dt>\n<dd id=\"fs-id2639302\">a null measurement device for measuring potentials (voltages)<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2447682\">\n<dt>ohmmeter<\/dt>\n<dd id=\"fs-id3103221\">an instrument that applies a voltage to a resistance, measures the current, calculates the resistance using Ohm\u2019s law, and provides a readout of this calculated resistance<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2391549\">\n<dt>bridge device<\/dt>\n<dd>a device that forms a bridge between two branches of a circuit; some bridge devices are used to make null measurements in circuits<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Wheatstone bridge<\/dt>\n<dd id=\"fs-id3285619\">a null measurement device for calculating resistance by balancing potential drops in a circuit<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1194","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1135,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1194","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1194\/revisions"}],"predecessor-version":[{"id":1195,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1194\/revisions\/1195"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1135"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1194\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1194"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1194"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1194"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1194"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}