{"id":1100,"date":"2017-10-27T16:31:24","date_gmt":"2017-10-27T16:31:24","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/resistance-and-resistivity\/"},"modified":"2017-11-08T03:26:15","modified_gmt":"2017-11-08T03:26:15","slug":"resistance-and-resistivity","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/resistance-and-resistivity\/","title":{"raw":"Resistance and Resistivity","rendered":"Resistance and Resistivity"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain the concept of resistivity.<\/li>\n<li>Use resistivity to calculate the resistance of specified configurations of material.<\/li>\n<li>Use the thermal coefficient of resistivity to calculate the change of resistance with temperature.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2990995\">\n<h1 data-type=\"title\">Material and Shape Dependence of Resistance<\/h1>\n<p id=\"import-auto-id1177114\">The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in <a href=\"#import-auto-id1351234\" class=\"autogenerated-content\">(Figure)<\/a> is easy to analyze, and, by so doing, we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder\u2019s electric resistance [latex]R[\/latex] is directly proportional to its length [latex]L[\/latex], similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again similar to the flow of fluid through a pipe). In fact, [latex]R[\/latex] is inversely proportional to the cylinder\u2019s cross-sectional area [latex]A[\/latex].<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1351234\">\n<div class=\"bc-figcaption figcaption\">A uniform cylinder of length [latex]L[\/latex] and cross-sectional area [latex]A[\/latex]. Its resistance to the flow of current is similar to the resistance posed by a pipe to fluid flow. The longer the cylinder, the greater its resistance. The larger its cross-sectional area [latex]A[\/latex], the smaller its resistance.<\/div>\n<p><span data-type=\"media\" data-alt=\"A cylindrical conductor of length L and cross section A is shown. The resistivity of the cylindrical section is represented as rho. The resistance of this cross section R is equal to rho L divided by A. The section of length L of cylindrical conductor is shown equivalent to a resistor represented by symbol R.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_21_03_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"A cylindrical conductor of length L and cross section A is shown. The resistivity of the cylindrical section is represented as rho. The resistance of this cross section R is equal to rho L divided by A. The section of length L of cylindrical conductor is shown equivalent to a resistor represented by symbol R.\" width=\"225\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2658962\">For a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the <span data-type=\"term\" id=\"import-auto-id2612960\">resistivity<\/span> [latex]\\rho [\/latex] of a substance so that the <strong data-effect=\"bold\">resistance<\/strong> [latex]R[\/latex] of an object is directly proportional to [latex]\\rho [\/latex]. Resistivity [latex]\\rho [\/latex] is an <em data-effect=\"italics\"><em data-effect=\"italics\">intrinsic <\/em><\/em> property of a material, independent of its shape or size. The resistance [latex]R[\/latex] of a uniform cylinder of length [latex]L[\/latex], of cross-sectional area [latex]A[\/latex], and made of a material with resistivity [latex]\\rho [\/latex], is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]R=\\frac{\\mathrm{\\rho L}}{A}\\text{.}[\/latex]<\/div>\n<p><a href=\"#import-auto-id1375921\" class=\"autogenerated-content\">(Figure)<\/a> gives representative values of [latex]\\rho [\/latex]. The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. Conductors have varying but large free charge densities, whereas most charges in insulators are bound to atoms and are not free to move. Semiconductors are intermediate, having far fewer free charges than conductors, but having properties that make the number of free charges depend strongly on the type and amount of impurities in the semiconductor. These unique properties of semiconductors are put to use in modern electronics, as will be explored in later chapters.<\/p>\n<table id=\"import-auto-id1375921\" summary=\"Table 21_03_01\">\n<caption><span data-type=\"title\">Resistivities        [latex]\\rho [\/latex]     of Various materials at       [latex]\\text{20\u00ba}\\text{C}[\/latex]     <\/span><\/caption>\n<thead>\n<tr>\n<th><span data-type=\"space\" data-count=\"2\">  <\/span>Material<\/th>\n<th>\n            Resistivity<br>\n              [latex]\\rho [\/latex]<br>\n              <strong data-effect=\"bold\"> (<\/strong><br>\n              [latex]\\Omega \\cdot \\text{m}[\/latex]<br>\n              <strong data-effect=\"bold\">)<\/strong>\n            <\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><em data-effect=\"italics\">Conductors<\/em><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silver<\/td>\n<td>[latex]1\\text{.}\\text{59}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Copper<\/td>\n<td>[latex]1\\text{.}\\text{72}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Gold<\/td>\n<td>[latex]2\\text{.}\\text{44}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Aluminum<\/td>\n<td>[latex]2\\text{.}\\text{65}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Tungsten<\/td>\n<td>[latex]5\\text{.}6\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Iron<\/td>\n<td>[latex]9\\text{.}\\text{71}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Platinum<\/td>\n<td>[latex]\\text{10}\\text{.}6\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Steel<\/td>\n<td>[latex]\\text{20}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Lead<\/td>\n<td>[latex]\\text{22}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Manganin (Cu, Mn, Ni alloy)<\/td>\n<td>[latex]\\text{44}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Constantan (Cu, Ni alloy)<\/td>\n<td>[latex]\\text{49}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Mercury<\/td>\n<td>[latex]\\text{96}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Nichrome (Ni, Fe, Cr alloy)<\/td>\n<td>[latex]\\text{100}\u00d7{\\text{10}}^{-8}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">Semiconductors<\/em><a data-type=\"footnote-number\" href=\"#footnote1\"><sup>1<\/sup><\/a><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Carbon (pure)<\/td>\n<td>[latex]\\text{3.5}\u00d7{\\text{10}}^{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Carbon<\/td>\n<td>[latex]\\left(3.5-\\text{60}\\right)\u00d7{\\text{10}}^{5}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Germanium (pure)<\/td>\n<td>[latex]\\text{600}\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Germanium<\/td>\n<td>[latex]\\left(1-\\text{600}\\right)\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silicon (pure)<\/td>\n<td>[latex]\\text{2300}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silicon<\/td>\n<td>[latex]\\text{0.1\u20132300}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">Insulators<\/em><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Amber<\/td>\n<td>[latex]5\u00d7{\\text{10}}^{\\text{14}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Glass<\/td>\n<td>[latex]{\\text{10}}^{9}-{\\text{10}}^{\\text{14}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Lucite<\/td>\n<td>[latex]{\\text{&gt;10}}^{\\text{13}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Mica<\/td>\n<td>[latex]{\\text{10}}^{\\text{11}}-{\\text{10}}^{\\text{15}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Quartz (fused)<\/td>\n<td>[latex]\\text{75}\u00d7{\\text{10}}^{\\text{16}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Rubber (hard)<\/td>\n<td>[latex]{\\text{10}}^{\\text{13}}-{\\text{10}}^{\\text{16}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Sulfur<\/td>\n<td>[latex]{\\text{10}}^{\\text{15}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Teflon<\/td>\n<td>[latex]{\\text{&gt;10}}^{\\text{13}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Wood<\/td>\n<td>[latex]{\\text{10}}^{8}-{\\text{10}}^{\\text{11}}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2056718\">\n<div data-type=\"title\" class=\"title\">Calculating Resistor Diameter: A Headlight Filament<\/div>\n<p>A car headlight filament is made of tungsten and has a cold resistance of [latex]0\\text{.}\\text{350}\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex]. If the filament is a cylinder 4.00 cm long (it may be coiled to save space), what is its diameter?<\/p>\n<p id=\"import-auto-id2991574\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2680858\">We can rearrange the equation [latex]R=\\frac{\\mathrm{\\rho L}}{A}[\/latex] to find the cross-sectional area [latex]A[\/latex] of the filament from the given information. Then its diameter can be found by assuming it has a circular cross-section.<\/p>\n<p id=\"import-auto-id1484936\"><strong>Solution<\/strong><\/p>\n<p>The cross-sectional area, found by rearranging the expression for the resistance of a cylinder given in [latex]R=\\frac{\\mathrm{\\rho L}}{A}[\/latex], is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]A=\\frac{\\mathrm{\\rho L}}{R}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1348601\">Substituting the given values, and taking [latex]\\rho [\/latex] from <a href=\"#import-auto-id1375921\" class=\"autogenerated-content\">(Figure)<\/a>, yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-614\">[latex]\\begin{array}{lll}A&amp; =&amp; \\frac{\\left(5.6\u00d7{\\text{10}}^{\u20138}\\phantom{\\rule{0.25em}{0ex}}\\Omega \\cdot \\text{m}\\right)\\left(4.00\u00d7{\\text{10}}^{\u20132}\\phantom{\\rule{0.25em}{0ex}}\\text{m}\\right)}{\\text{0.350}\\phantom{\\rule{0.25em}{0ex}}\\Omega }\\\\ &amp; =&amp; \\text{6.40}\u00d7{\\text{10}}^{\u20139}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1327496\">The area of a circle is related to its diameter [latex]D[\/latex] by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-83\">[latex]A=\\frac{{\\mathrm{\\pi D}}^{2}}{4}\\text{.}[\/latex]<\/div>\n<p>Solving for the diameter [latex]D[\/latex], and substituting the value found for [latex]A[\/latex], gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}D&amp; =&amp; \\text{2}{\\left(\\frac{A}{p}\\right)}^{\\frac{1}{2}}=\\text{2}{\\left(\\frac{6.40\u00d7{\\text{10}}^{\u20139}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{2}}{3.14}\\right)}^{\\frac{1}{2}}\\\\ &amp; =&amp; 9.0\u00d7{\\text{10}}^{\u20135}\\phantom{\\rule{0.25em}{0ex}}\\text{m}\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id1981021\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2950520\">The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because [latex]\\rho [\/latex] is known to only two digits.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3035815\">\n<h1 data-type=\"title\">Temperature Variation of Resistance<\/h1>\n<p id=\"import-auto-id2578283\">The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See <a href=\"#import-auto-id3201924\" class=\"autogenerated-content\">(Figure)<\/a>.) Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal make more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about [latex]\\text{100\u00ba}\\text{C}[\/latex] or less), resistivity [latex]\\rho [\/latex] varies with temperature change [latex]\\Delta T[\/latex] as expressed in the following equation<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\rho ={\\rho }_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)\\text{,}[\/latex]<\/div>\n<p id=\"import-auto-id1872967\">where [latex]{\\rho }_{0}[\/latex] is the original resistivity and [latex]\\alpha [\/latex] is the <span data-type=\"term\" id=\"import-auto-id2990408\">temperature coefficient of resistivity<\/span>. (See the values of [latex]\\alpha [\/latex] in <a href=\"#import-auto-id1382426\" class=\"autogenerated-content\">(Figure)<\/a> below.) For larger temperature changes, [latex]\\alpha [\/latex] may vary or a nonlinear equation may be needed to find [latex]\\rho [\/latex]. Note that [latex]\\alpha [\/latex] is positive for metals, meaning their resistivity increases with temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (which is made of copper, manganese and nickel), for example, has [latex]\\alpha [\/latex]<em data-effect=\"italics\"> close to zero (to three digits on the scale in <a href=\"#import-auto-id1382426\" class=\"autogenerated-content\">(Figure)<\/a>), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.<\/em><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3201924\">\n<div class=\"bc-figcaption figcaption\">The resistance of a sample of mercury is zero at very low temperatures\u2014it is a superconductor up to about 4.2 K. Above that critical temperature, its resistance makes a sudden jump and then increases nearly linearly with temperature.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1374087\" data-alt=\"A graph for variation of resistance R with temperature T for a mercury sample is shown. The temperature T is plotted along the x axis and is measured in Kelvin, and the resistance R is plotted along the y axis and is measured in ohms. The curve starts at x equals zero and y equals zero, and coincides with the X axis until the value of temperature is four point two Kelvin, known as the critical temperature T sub c. At temperature T sub c, the curve shows a vertical rise, represented by a dotted line, until the resistance is about zero point one one ohms. After this temperature the resistance shows a nearly linear increase with temperature T.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_21_03_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"A graph for variation of resistance R with temperature T for a mercury sample is shown. The temperature T is plotted along the x axis and is measured in Kelvin, and the resistance R is plotted along the y axis and is measured in ohms. The curve starts at x equals zero and y equals zero, and coincides with the X axis until the value of temperature is four point two Kelvin, known as the critical temperature T sub c. At temperature T sub c, the curve shows a vertical rise, represented by a dotted line, until the resistance is about zero point one one ohms. After this temperature the resistance shows a nearly linear increase with temperature T.\" width=\"200\"><\/span><\/p><\/div>\n<table summary=\"Table 21_03_02\">\n<caption><span data-type=\"title\">Tempature Coefficients of Resistivity   [latex]\\alpha [\/latex]     <\/span><\/caption>\n<thead>\n<tr>\n<th><span data-type=\"space\" data-count=\"2\">  <\/span>Material<\/th>\n<th>Coefficient [latex]\\alpha [\/latex]<em data-effect=\"italics\">(1\/\u00b0C)<a data-type=\"footnote-number\" href=\"#footnote2\"><sup>2<\/sup><\/a><\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><em data-effect=\"italics\">Conductors<\/em><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silver<\/td>\n<td>[latex]3\\text{.}8\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Copper<\/td>\n<td>[latex]3\\text{.}9\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Gold<\/td>\n<td>[latex]3\\text{.}4\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Aluminum<\/td>\n<td>[latex]3\\text{.}9\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Tungsten<\/td>\n<td>[latex]4\\text{.}5\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Iron<\/td>\n<td>[latex]5\\text{.}0\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Platinum<\/td>\n<td>[latex]3\\text{.}\\text{93}\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Lead<\/td>\n<td>[latex]3\\text{.}9\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Manganin (Cu, Mn, Ni alloy)<\/td>\n<td>[latex]0\\text{.}\\text{000}\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Constantan (Cu, Ni alloy)<\/td>\n<td>[latex]0\\text{.}\\text{002}\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Mercury<\/td>\n<td>[latex]0\\text{.}\\text{89}\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Nichrome (Ni, Fe, Cr alloy)<\/td>\n<td>[latex]0\\text{.}4\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">Semiconductors<\/em><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Carbon (pure)<\/td>\n<td>[latex]-0\\text{.}5\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Germanium (pure)<\/td>\n<td>[latex]-\\text{50}\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silicon (pure)<\/td>\n<td>[latex]-\\text{70}\u00d7{\\text{10}}^{-3}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id2973346\">Note also that [latex]\\alpha [\/latex] is negative for the semiconductors listed in <a href=\"#import-auto-id1382426\" class=\"autogenerated-content\">(Figure)<\/a>, meaning that their resistivity decreases with increasing temperature. They become better conductors at higher temperature, because increased thermal agitation increases the number of free charges available to carry current. This property of decreasing [latex]\\rho [\/latex] with temperature is also related to the type and amount of impurities present in the semiconductors.<\/p>\n<p id=\"import-auto-id2589471\">The resistance of an object also depends on temperature, since [latex]{R}_{0}[\/latex] is directly proportional to [latex]\\rho [\/latex]. For a cylinder we know [latex]R=\\mathrm{\\rho L}\/A[\/latex], and so, if [latex]L[\/latex] and [latex]A[\/latex] do not change greatly with temperature, [latex]R[\/latex] will have the same temperature dependence as [latex]\\rho [\/latex]. (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on [latex]L[\/latex] and [latex]A[\/latex] is about two orders of magnitude less than on [latex]\\rho [\/latex].) Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-145\">[latex]R={R}_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)[\/latex]<\/div>\n<p id=\"import-auto-id2980135\">is the temperature dependence of the resistance of an object, where [latex]{R}_{0}[\/latex] is the original resistance and [latex]R[\/latex] is the resistance after a temperature change [latex]\\Delta T[\/latex]. Numerous thermometers are based on the effect of temperature on resistance. (See <a href=\"#import-auto-id1568367\" class=\"autogenerated-content\">(Figure)<\/a>.) One of the most common is the thermistor, a semiconductor crystal with a strong temperature dependence, the resistance of which is measured to obtain its temperature. The device is small, so that it quickly comes into thermal equilibrium with the part of a person it touches.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1568367\">\n<div class=\"bc-figcaption figcaption\">These familiar thermometers are based on the automated measurement of a thermistor\u2019s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2594606\" data-alt=\"A photograph showing two digital thermometers used for measuring body temperature.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_21_03_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"A photograph showing two digital thermometers used for measuring body temperature.\" width=\"250\"><\/span><\/p><\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1889050\">\n<div data-type=\"title\" class=\"title\">Calculating Resistance: Hot-Filament Resistance<\/div>\n<p id=\"import-auto-id3013135\">Although caution must be used in applying [latex]\\rho ={\\rho }_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)[\/latex] and [latex]R={R}_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)[\/latex] for temperature changes greater than [latex]\\text{100\u00ba}\\text{C}[\/latex], for tungsten the equations work reasonably well for very large temperature changes. What, then, is the resistance of the tungsten filament in the previous example if its temperature is increased from room temperature ([latex]\\text{20\u00baC}[\/latex]) to a typical operating temperature of [latex]\\text{2850\u00ba}\\text{C}[\/latex]?<\/p>\n<p id=\"import-auto-id2057367\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1997770\">This is a straightforward application of [latex]R={R}_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)[\/latex], since the original resistance of the filament was given to be [latex]{R}_{0}=0\\text{.}\\text{350 \u03a9}[\/latex], and the temperature change is [latex]\\Delta T=\\text{2830\u00ba}\\text{C}[\/latex].<\/p>\n<p id=\"import-auto-id2670063\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1416811\">The hot resistance [latex]R[\/latex] is obtained by entering known values into the above equation:<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{lll}R&amp; =&amp; {R}_{0}\\left(1+\\alpha \\Delta T\\right)\\\\ &amp; =&amp; \\left(0\\text{.}\\text{350 \u03a9}\\right)\\left[\\text{1}+\\left(4.5\u00d7{\\text{10}}^{\u20133}\/\\text{\u00baC}\\right)\\left(\\text{2830\u00ba}\\text{C}\\right)\\right]\\\\ &amp; =&amp; \\text{4.8 \u03a9.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3054482\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2625691\">This value is consistent with the headlight resistance example in <a href=\"\/contents\/c8b9a2f8-cb8f-484f-b53c-72459218c189@4\">Ohm\u2019s Law: Resistance and Simple Circuits<\/a>.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Resistance in a Wire<\/div>\n<p>Learn about the physics of resistance in a wire. Change its resistivity, length, and area to see how they affect the wire's resistance. The sizes of the symbols in the equation change along with the diagram of a wire.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id2470478\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/c55ee679dd7c4862e0e67fdc8fa3b8eb275eea29\/resistance-in-a-wire_en.jar\">Resistance in a Wire<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_21.3\" data-alt=\"\"><a href=\"\/resources\/c55ee679dd7c4862e0e67fdc8fa3b8eb275eea29\/resistance-in-a-wire_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/jpg\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/jpg\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id2423329\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id3176599\">\n<li id=\"import-auto-id3017858\">The resistance [latex]R[\/latex] of a cylinder of length [latex]L[\/latex] and cross-sectional area [latex]A[\/latex] is [latex]R=\\frac{\\mathrm{\\rho L}}{A}[\/latex], where [latex]\\rho [\/latex] is the resistivity of the material.<\/li>\n<li id=\"import-auto-id2684104\">Values of [latex]\\rho [\/latex] in <a href=\"#import-auto-id1375921\" class=\"autogenerated-content\">(Figure)<\/a> show that materials fall into three groups\u2014<em data-effect=\"italics\">conductors, semiconductors, and insulators<\/em>.<\/li>\n<li id=\"import-auto-id3008174\">Temperature affects resistivity; for relatively small temperature changes [latex]\\Delta T[\/latex], resistivity is [latex]\\rho ={\\rho }_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)[\/latex], where [latex]{\\rho }_{0}[\/latex] is the original resistivity and [latex]\\text{\u03b1}[\/latex]  is the temperature coefficient of resistivity.<\/li>\n<li><a href=\"#import-auto-id1382426\" class=\"autogenerated-content\">(Figure)<\/a> gives values for [latex]\\alpha [\/latex], the temperature coefficient of resistivity.<\/li>\n<li id=\"import-auto-id1609529\">The resistance [latex]R[\/latex] of an object also varies with temperature: [latex]R={R}_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)[\/latex], where [latex]{R}_{0}[\/latex] is the original resistance, and [latex]R[\/latex]  is the resistance after the temperature change.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1994296\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2446582\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2680317\">\n<p id=\"import-auto-id3224512\">In which of the three semiconducting materials listed in <a href=\"#import-auto-id1375921\" class=\"autogenerated-content\">(Figure)<\/a> do impurities supply free charges? (Hint: Examine the range of resistivity for each and determine whether the pure semiconductor has the higher or lower conductivity.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1575560\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3051109\">\n<p id=\"import-auto-id2684990\">Does the resistance of an object depend on the path current takes through it? Consider, for example, a rectangular bar\u2014is its resistance the same along its length as across its width? (See <a href=\"#import-auto-id1435027\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1435027\">\n<div class=\"bc-figcaption figcaption\">Does current taking two different paths through the same object encounter different resistance?<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1576396\" data-alt=\"Part a of the figure shows a voltage V applied along the length of a rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the length of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the length is shown as R and the current is shown as I. Part b of the figure shows a voltage V applied along the width of the same rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the width of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the width is shown as R prime, and the current is shown as I prime.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_21_03_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows a voltage V applied along the length of a rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the length of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the length is shown as R and the current is shown as I. Part b of the figure shows a voltage V applied along the width of the same rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the width of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the width is shown as R prime, and the current is shown as I prime.\" width=\"325\"><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2452298\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1120880\">\n<p id=\"import-auto-id3105586\">If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1404656\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3201532\">\n<p id=\"import-auto-id1047618\">Explain why [latex]R={R}_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)[\/latex] for the temperature variation of the resistance [latex]R[\/latex] of an object is not as accurate as [latex]\\rho ={\\rho }_{0}\\left(\\text{1}+\\alpha \\Delta T\\right)[\/latex], which gives the temperature variation of resistivity [latex]\\rho [\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1815817\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2442974\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2930887\">\n<p id=\"import-auto-id1441418\">What is the resistance of a 20.0-m-long piece of 12-gauge copper wire having a 2.053-mm diameter?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1221864\">\n<p id=\"import-auto-id3423212\">[latex]\\text{0.104 \u03a9}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1871839\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2639671\">\n<p id=\"import-auto-id1525157\">The diameter of 0-gauge copper wire is 8.252 mm. Find the resistance of a 1.00-km length of such wire used for power transmission.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2668793\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1932262\">\n<p id=\"import-auto-id3017926\">If the 0.100-mm diameter tungsten filament in a light bulb is to have a resistance of<br>\n[latex]\\text{0.200 \u03a9}[\/latex]<br>\nat<br>\n[latex]\\text{20}\\text{.}0\u00ba\\text{C}[\/latex], how long should it be?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1429348\">\n<p id=\"import-auto-id3424736\">[latex]2\\text{.}\\text{8}\u00d7{\\text{10}}^{-2}\\phantom{\\rule{0.25em}{0ex}}\\text{m}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3286438\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2454151\">\n<p id=\"import-auto-id3250821\">Find the ratio of the diameter of aluminum to copper wire, if they have the same resistance per unit length (as they might in household wiring).<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2396605\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3358944\">\n<p id=\"import-auto-id360022\">What current flows through a 2.54-cm-diameter rod of pure silicon that is 20.0 cm long, when<br>\n[latex]{1.00 \u00d7 10}^{\\text{3}}\\phantom{\\rule{0.25em}{0ex}}\\text{V}[\/latex]<br>\n is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3034098\">\n<p id=\"import-auto-id3173375\">[latex]1\\text{.}\\text{10}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{A}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3402655\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2599616\">\n<p id=\"import-auto-id934566\">(a) To what temperature must you raise a copper wire, originally at<br>\n[latex]\\text{20.0\u00baC}[\/latex],<br>\nto double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances?<\/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-id2691474\">\n<p id=\"import-auto-id1447169\">A resistor made of Nichrome wire is used in an application where its resistance cannot change more than 1.00% from its value at [latex]\\text{20}\\text{.}0\u00ba\\text{C}[\/latex]. Over what temperature range can it be used?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2514795\">\n<p id=\"import-auto-id1934170\">[latex]-5\u00ba\\text{C to 45\u00baC}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3449589\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2589289\">\n<p id=\"import-auto-id1220869\">Of what material is a resistor made if its resistance is 40.0% greater at [latex]\\text{100\u00ba}\\text{C}[\/latex] than at [latex]\\text{20}\\text{.}0\u00ba\\text{C}[\/latex]?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3201853\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3085520\">\n<p id=\"import-auto-id1374451\">An electronic device designed to operate at any temperature in the range from [latex]\\text{\u201310}\\text{.}0\u00ba\\text{C to 55}\\text{.}0\u00ba\\text{C}[\/latex] contains pure carbon resistors. By what factor does their resistance increase over this range?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1917734\">\n<p id=\"import-auto-id1868093\">1.03<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1973609\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3385545\">\n<p id=\"import-auto-id1526233\">(a) Of what material is a wire made, if it is 25.0 m long with a 0.100 mm diameter and has a resistance of [latex]\\text{77}\\text{.}7\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex] at [latex]\\text{20}\\text{.}0\u00ba\\text{C}[\/latex]? (b) What is its resistance at [latex]\\text{150\u00ba}\\text{C}[\/latex]?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3103900\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3454882\">\n<p id=\"import-auto-id3177308\">Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at [latex]\\text{20}\\text{.}0\u00ba\\text{C}[\/latex]?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3016931\">\n<p id=\"import-auto-id2956468\">0.06%<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3110369\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3044201\">\n<p id=\"import-auto-id3094331\">A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2448773\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2653584\">\n<p id=\"import-auto-id2454057\">A copper wire has a resistance of [latex]0\\text{.}\\text{500}\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex] at [latex]\\text{20}\\text{.}0\u00ba\\text{C}[\/latex], and an iron wire has a resistance of [latex]0\\text{.}\\text{525}\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex] at the same temperature. At what temperature are their resistances equal?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1998215\">\n<p id=\"import-auto-id2615498\">[latex]-\\text{17\u00ba}\\text{C}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2382175\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3201469\">\n<p id=\"import-auto-id1560690\">(a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has [latex]\\alpha =\u20130\\text{.}\\text{0600}\/\\text{\u00baC}[\/latex]) when it is at the same temperature as the patient. What is a patient\u2019s temperature if the thermistor\u2019s resistance at that temperature is 82.0% of its value at [latex]\\text{37}\\text{.}0\u00ba\\text{C}[\/latex] (normal body temperature)? (b) The negative value<br>\nfor [latex]\\alpha [\/latex]<em data-effect=\"italics\"> may not be maintained for very low temperatures. Discuss why and whether this is the case here. (Hint: Resistance can\u2019t become negative.)<\/em><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3189568\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1419790\">\n<p id=\"import-auto-id3388738\"><strong>Integrated Concepts<\/strong><\/p>\n<p id=\"eip-id2581512\">(a) Redo <a href=\"#fs-id1871839\" class=\"autogenerated-content\">(Figure)<\/a> taking into account the thermal expansion of the tungsten filament. You may assume a thermal expansion coefficient of [latex]\\text{12}\u00d7{\\text{10}}^{-6}\/\\text{\u00baC}[\/latex]. (b) By what percentage does your answer differ from that in the example?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1599741\">\n<p id=\"import-auto-id2963593\">(a) [latex]4\\text{.}7\\phantom{\\rule{0.25em}{0ex}}\\Omega [\/latex] (total)<\/p>\n<p id=\"import-auto-id2930991\">(b) 3.0% decrease<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2415615\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1842428\">\n<p id=\"import-auto-id3090921\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"eip-id1546986\">(a) To what temperature must you raise a resistor made of constantan to double its resistance, assuming a constant temperature coefficient of resistivity? (b) To cut it in half? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable, or which premises are inconsistent?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"footnote-refs\">\n<h2 data-type=\"footnote-title\">Footnotes<\/h2>\n<ol>\n<li><a data-type=\"footnote-ref\" href=\"#footnote-ref1\">1<\/a> Values depend strongly on amounts and types of impurities<\/li>\n<li><a data-type=\"footnote-ref\" href=\"#footnote-ref2\">2<\/a> Values at  20\u00b0C.<\/li>\n<\/ol>\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-id1012671\">\n<dt>resistivity<\/dt>\n<dd id=\"fs-id3082005\">an intrinsic property of a material, independent of its shape or size, directly proportional to the resistance, denoted by <em data-effect=\"italics\">\u03c1<\/em><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3305400\">\n<dt>temperature coefficient of resistivity<\/dt>\n<dd id=\"fs-id2448387\">an empirical quantity, denoted by <em data-effect=\"italics\">\u03b1<\/em>, which describes the change in resistance or resistivity of a material with temperature<\/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 the concept of resistivity.<\/li>\n<li>Use resistivity to calculate the resistance of specified configurations of material.<\/li>\n<li>Use the thermal coefficient of resistivity to calculate the change of resistance with temperature.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2990995\">\n<h1 data-type=\"title\">Material and Shape Dependence of Resistance<\/h1>\n<p id=\"import-auto-id1177114\">The resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in <a href=\"#import-auto-id1351234\" class=\"autogenerated-content\">(Figure)<\/a> is easy to analyze, and, by so doing, we can gain insight into the resistance of more complicated shapes. As you might expect, the cylinder\u2019s electric resistance <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 directly proportional to its length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, similar to the resistance of a pipe to fluid flow. The longer the cylinder, the more collisions charges will make with its atoms. The greater the diameter of the cylinder, the more current it can carry (again similar to the flow of fluid through a pipe). In fact, <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 inversely proportional to the cylinder\u2019s cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1351234\">\n<div class=\"bc-figcaption figcaption\">A uniform cylinder of length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> and cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>. Its resistance to the flow of current is similar to the resistance posed by a pipe to fluid flow. The longer the cylinder, the greater its resistance. The larger its cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>, the smaller its resistance.<\/div>\n<p><span data-type=\"media\" data-alt=\"A cylindrical conductor of length L and cross section A is shown. The resistivity of the cylindrical section is represented as rho. The resistance of this cross section R is equal to rho L divided by A. The section of length L of cylindrical conductor is shown equivalent to a resistor represented by symbol R.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_21_03_01a.jpg\" data-media-type=\"image\/jpg\" alt=\"A cylindrical conductor of length L and cross section A is shown. The resistivity of the cylindrical section is represented as rho. The resistance of this cross section R is equal to rho L divided by A. The section of length L of cylindrical conductor is shown equivalent to a resistor represented by symbol R.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2658962\">For a given shape, the resistance depends on the material of which the object is composed. Different materials offer different resistance to the flow of charge. We define the <span data-type=\"term\" id=\"import-auto-id2612960\">resistivity<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> of a substance so that the <strong data-effect=\"bold\">resistance<\/strong> <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;\" \/> of an object is directly proportional to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. Resistivity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> is an <em data-effect=\"italics\"><em data-effect=\"italics\">intrinsic <\/em><\/em> property of a material, independent of its shape or size. The resistance <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;\" \/> of a uniform cylinder of length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/>, of cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>, and made of a material with resistivity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>, is<\/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-5cf9b9e8f06e9d0984af2e1c550fd28d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#114;&#104;&#111;&#32;&#76;&#125;&#125;&#123;&#65;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"61\" style=\"vertical-align: -6px;\" \/><\/div>\n<p><a href=\"#import-auto-id1375921\" class=\"autogenerated-content\">(Figure)<\/a> gives representative values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. The materials listed in the table are separated into categories of conductors, semiconductors, and insulators, based on broad groupings of resistivities. Conductors have the smallest resistivities, and insulators have the largest; semiconductors have intermediate resistivities. Conductors have varying but large free charge densities, whereas most charges in insulators are bound to atoms and are not free to move. Semiconductors are intermediate, having far fewer free charges than conductors, but having properties that make the number of free charges depend strongly on the type and amount of impurities in the semiconductor. These unique properties of semiconductors are put to use in modern electronics, as will be explored in later chapters.<\/p>\n<table id=\"import-auto-id1375921\" summary=\"Table 21_03_01\">\n<caption><span data-type=\"title\">Resistivities        <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>     of Various materials at       <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5999fba056332fa4b6462b4ad19723a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/>     <\/span><\/caption>\n<thead>\n<tr>\n<th><span data-type=\"space\" data-count=\"2\">  <\/span>Material<\/th>\n<th>\n            Resistivity<br \/>\n              <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/><br \/>\n              <strong data-effect=\"bold\"> (<\/strong><br \/>\n              <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-732cd8de7971a1d88ebec373d36dd828_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#79;&#109;&#101;&#103;&#97;&#32;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><br \/>\n              <strong data-effect=\"bold\">)<\/strong>\n            <\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><em data-effect=\"italics\">Conductors<\/em><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silver<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f16487019106a95cfae031ab0f67ff0_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;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Copper<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1d4a2be4f2206742c80db9465f63c3cd_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;&#55;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Gold<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-160f52734b43cbf28c596a3a1070fcfe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Aluminum<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a068b56d625e0d5aa126f9377d9e53c3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Tungsten<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2e7c2f96dd670ef9b9a02b69462cf13b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#54;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Iron<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-16856b14f85643adcdc85a72af95382f_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;&#55;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Platinum<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-33d1c74b81ef888bca7caa3f13b3a608_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;&#54;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Steel<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-03782e5e7583911ed48383ab7b631cb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Lead<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ea25cff5b2df33658cd692cce4396cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Manganin (Cu, Mn, Ni alloy)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b3d9da761211b54ae35d3f989714bbdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Constantan (Cu, Ni alloy)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6634fc4ba5296ecc5fe1f86ddd250e86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Mercury<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-04c5aff3c8d24d1576cc747952ad4be8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"53\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Nichrome (Ni, Fe, Cr alloy)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-90220e69421b181729e039dbbece91e3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#56;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">Semiconductors<\/em><a data-type=\"footnote-number\" href=\"#footnote1\"><sup>1<\/sup><\/a><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Carbon (pure)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f188da96140c8a0755e6af107e5a5cec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#46;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Carbon<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d727420ee891cd157900aae19aabd1d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#46;&#53;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Germanium (pure)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9dad8b2b6bfcf2e382a4cbcd68bf662f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"62\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Germanium<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2936e9e29a7dcc4ae2453065ac812b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#48;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"109\" style=\"vertical-align: -4px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silicon (pure)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2332392f4b835df205f593abb8be98e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"36\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silicon<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8cdca11d787c0f455816769212c594a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#49;&#45;&#50;&#51;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"65\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">Insulators<\/em><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Amber<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d97d6e3df0e385e60e35323e50ded76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Glass<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d036da3679050c345f9fef9edb923ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#57;&#125;&#45;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Lucite<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-96c9cdc4886b1dbfaddc900c8b884c8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#62;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Mica<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d8f27b41cfde449ce411539137aea29f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&#125;&#45;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Quartz (fused)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c198bafe3eba1e3b249624b97ad83ab1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"49\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Rubber (hard)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-277c729acd972e755a5a58eee41cd7bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;&#45;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Sulfur<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e3f95db434a3c915e4c9a8a87790a325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"31\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Teflon<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-96c9cdc4886b1dbfaddc900c8b884c8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#62;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"40\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Wood<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4c4361f157155b0d599733b346c93d2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#45;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2056718\">\n<div data-type=\"title\" class=\"title\">Calculating Resistor Diameter: A Headlight Filament<\/div>\n<p>A car headlight filament is made of tungsten and has a cold resistance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-44df047ea71b6c46a8680bd682dc76ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#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=\"57\" style=\"vertical-align: 0px;\" \/>. If the filament is a cylinder 4.00 cm long (it may be coiled to save space), what is its diameter?<\/p>\n<p id=\"import-auto-id2991574\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2680858\">We can rearrange the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdb8da839bea2886124a1aa958d7444f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#114;&#104;&#111;&#32;&#76;&#125;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"55\" style=\"vertical-align: -6px;\" \/> to find the cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> of the filament from the given information. Then its diameter can be found by assuming it has a circular cross-section.<\/p>\n<p id=\"import-auto-id1484936\"><strong>Solution<\/strong><\/p>\n<p>The cross-sectional area, found by rearranging the expression for the resistance of a cylinder given in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdb8da839bea2886124a1aa958d7444f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#114;&#104;&#111;&#32;&#76;&#125;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"55\" style=\"vertical-align: -6px;\" \/>, is<\/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-058b3879e815434f96901b0e9a54b3ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#114;&#104;&#111;&#32;&#76;&#125;&#125;&#123;&#82;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"import-auto-id1348601\">Substituting the given values, and taking <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> from <a href=\"#import-auto-id1375921\" class=\"autogenerated-content\">(Figure)<\/a>, yields<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-614\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c6fdf101b2ca11093f0c49b459feee26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#65;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#46;&#54;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#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;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#50;&#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;&#109;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#51;&#53;&#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;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#46;&#52;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"47\" width=\"237\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"import-auto-id1327496\">The area of a circle is related to its diameter <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/> by<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-83\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6e2b7c1edfc584ecfa22b54379005faa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#112;&#105;&#32;&#68;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"70\" style=\"vertical-align: -6px;\" \/><\/div>\n<p>Solving for the diameter <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4b9ef1bbd23fd1b198de883813285620_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#68;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/>, and substituting the value found for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>, 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-6d0f3ad2395e8bcfd1a8cd0027be26ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#68;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#65;&#125;&#123;&#112;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#46;&#52;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#57;&#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;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#125;&#123;&#51;&#46;&#49;&#52;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#125;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#57;&#46;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#53;&#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;&#109;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"54\" width=\"259\" style=\"vertical-align: -20px;\" \/><\/div>\n<p id=\"import-auto-id1981021\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2950520\">The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> is known to only two digits.<\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id3035815\">\n<h1 data-type=\"title\">Temperature Variation of Resistance<\/h1>\n<p id=\"import-auto-id2578283\">The resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See <a href=\"#import-auto-id3201924\" class=\"autogenerated-content\">(Figure)<\/a>.) Conversely, the resistivity of conductors increases with increasing temperature. Since the atoms vibrate more rapidly and over larger distances at higher temperatures, the electrons moving through a metal make more collisions, effectively making the resistivity higher. Over relatively small temperature changes (about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95453086a14fb8fcafc6b2bcfedcb280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"38\" style=\"vertical-align: -1px;\" \/> or less), resistivity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> varies with temperature change <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cb5a269f1ca84874583e8696a1b59bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/> as expressed in the following equation<\/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-f07b45fa87b0ced0db5debbf0b8df1a8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;&#61;&#123;&#92;&#114;&#104;&#111;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"143\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1872967\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0fed00b387c11493e556843dd3ce1ab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#114;&#104;&#111;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/> is the original resistivity and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is the <span data-type=\"term\" id=\"import-auto-id2990408\">temperature coefficient of resistivity<\/span>. (See the values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> in <a href=\"#import-auto-id1382426\" class=\"autogenerated-content\">(Figure)<\/a> below.) For larger temperature changes, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> may vary or a nonlinear equation may be needed to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is positive for metals, meaning their resistivity increases with temperature. Some alloys have been developed specifically to have a small temperature dependence. Manganin (which is made of copper, manganese and nickel), for example, has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\"> close to zero (to three digits on the scale in <a href=\"#import-auto-id1382426\" class=\"autogenerated-content\">(Figure)<\/a>), and so its resistivity varies only slightly with temperature. This is useful for making a temperature-independent resistance standard, for example.<\/em><\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id3201924\">\n<div class=\"bc-figcaption figcaption\">The resistance of a sample of mercury is zero at very low temperatures\u2014it is a superconductor up to about 4.2 K. Above that critical temperature, its resistance makes a sudden jump and then increases nearly linearly with temperature.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1374087\" data-alt=\"A graph for variation of resistance R with temperature T for a mercury sample is shown. The temperature T is plotted along the x axis and is measured in Kelvin, and the resistance R is plotted along the y axis and is measured in ohms. The curve starts at x equals zero and y equals zero, and coincides with the X axis until the value of temperature is four point two Kelvin, known as the critical temperature T sub c. At temperature T sub c, the curve shows a vertical rise, represented by a dotted line, until the resistance is about zero point one one ohms. After this temperature the resistance shows a nearly linear increase with temperature T.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_21_03_02a.jpg\" data-media-type=\"image\/jpg\" alt=\"A graph for variation of resistance R with temperature T for a mercury sample is shown. The temperature T is plotted along the x axis and is measured in Kelvin, and the resistance R is plotted along the y axis and is measured in ohms. The curve starts at x equals zero and y equals zero, and coincides with the X axis until the value of temperature is four point two Kelvin, known as the critical temperature T sub c. At temperature T sub c, the curve shows a vertical rise, represented by a dotted line, until the resistance is about zero point one one ohms. After this temperature the resistance shows a nearly linear increase with temperature T.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<table summary=\"Table 21_03_02\">\n<caption><span data-type=\"title\">Tempature Coefficients of Resistivity   <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>     <\/span><\/caption>\n<thead>\n<tr>\n<th><span data-type=\"space\" data-count=\"2\">  <\/span>Material<\/th>\n<th>Coefficient <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\">(1\/\u00b0C)<a data-type=\"footnote-number\" href=\"#footnote2\"><sup>2<\/sup><\/a><\/em><\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><em data-effect=\"italics\">Conductors<\/em><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silver<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8521e5e6a1f06df82eac74e5300a48df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#56;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Copper<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ed866b3149a863f1e4ff4948d997636_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#57;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Gold<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ee740cb3acabf47ef9ef4569208a2944_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Aluminum<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ed866b3149a863f1e4ff4948d997636_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#57;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Tungsten<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-15898625a218f66b835181484472887e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Iron<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f8edd8d7fd928254d2bbf33166112641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Platinum<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1d223b85ff806d8c88ace38dcb808f9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#51;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Lead<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6ed866b3149a863f1e4ff4948d997636_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#57;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Manganin (Cu, Mn, Ni alloy)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-da5eacce706d3bbe6ce2f6cf66295104_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Constantan (Cu, Ni alloy)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9c27f5158867b9d6b4a2dc6b5640360_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"76\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Mercury<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a929dece9898338c56e8623e6f499521_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#57;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"67\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Nichrome (Ni, Fe, Cr alloy)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-16d19a6a7b3aadd7043c68d3fc7e281e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><em data-effect=\"italics\">Semiconductors<\/em><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Carbon (pure)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-87be95c4a57b8e057fec092e102f5085_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"71\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Germanium (pure)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-110df451c34347649e6dbdfb301638aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><span data-type=\"space\" data-count=\"2\">  <\/span>Silicon (pure)<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f09c8932efd7a2b5763a899a2d505580_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -1px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"import-auto-id2973346\">Note also that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is negative for the semiconductors listed in <a href=\"#import-auto-id1382426\" class=\"autogenerated-content\">(Figure)<\/a>, meaning that their resistivity decreases with increasing temperature. They become better conductors at higher temperature, because increased thermal agitation increases the number of free charges available to carry current. This property of decreasing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> with temperature is also related to the type and amount of impurities present in the semiconductors.<\/p>\n<p id=\"import-auto-id2589471\">The resistance of an object also depends on temperature, since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-522741a1a3befc8cbda2a31eded25676_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> is directly proportional to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. For a cylinder we know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4989fc0e1c40a2e7c275d35c5ea70817_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#114;&#104;&#111;&#32;&#76;&#125;&#47;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"78\" style=\"vertical-align: -5px;\" \/>, and so, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> do not change greatly with temperature, <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;\" \/> will have the same temperature dependence as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>. (Examination of the coefficients of linear expansion shows them to be about two orders of magnitude less than typical temperature coefficients of resistivity, and so the effect of temperature on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> is about two orders of magnitude less than on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>.) Thus,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"eip-145\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-356822f8c20092ce207f835adb4a95f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#123;&#82;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id2980135\">is the temperature dependence of the resistance of an object, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-522741a1a3befc8cbda2a31eded25676_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> is the original resistance and <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 the resistance after a temperature change <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cb5a269f1ca84874583e8696a1b59bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/>. Numerous thermometers are based on the effect of temperature on resistance. (See <a href=\"#import-auto-id1568367\" class=\"autogenerated-content\">(Figure)<\/a>.) One of the most common is the thermistor, a semiconductor crystal with a strong temperature dependence, the resistance of which is measured to obtain its temperature. The device is small, so that it quickly comes into thermal equilibrium with the part of a person it touches.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1568367\">\n<div class=\"bc-figcaption figcaption\">These familiar thermometers are based on the automated measurement of a thermistor\u2019s temperature-dependent resistance. (credit: Biol, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2594606\" data-alt=\"A photograph showing two digital thermometers used for measuring body temperature.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_21_03_03a.jpg\" data-media-type=\"image\/jpg\" alt=\"A photograph showing two digital thermometers used for measuring body temperature.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1889050\">\n<div data-type=\"title\" class=\"title\">Calculating Resistance: Hot-Filament Resistance<\/div>\n<p id=\"import-auto-id3013135\">Although caution must be used in applying <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3585d03aacf53b997ff1e5284e7772ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;&#61;&#123;&#92;&#114;&#104;&#111;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-356822f8c20092ce207f835adb4a95f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#123;&#82;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/> for temperature changes greater than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95453086a14fb8fcafc6b2bcfedcb280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"38\" style=\"vertical-align: -1px;\" \/>, for tungsten the equations work reasonably well for very large temperature changes. What, then, is the resistance of the tungsten filament in the previous example if its temperature is increased from room temperature (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8c5021f856100c9924f694724df9862a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/>) to a typical operating temperature of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a47be6d7070e306f7da2daefe79193fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#56;&#53;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: 0px;\" \/>?<\/p>\n<p id=\"import-auto-id2057367\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id1997770\">This is a straightforward application of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-356822f8c20092ce207f835adb4a95f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#123;&#82;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/>, since the original resistance of the filament was given to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-75c27f573378758f7f0001c036af4b56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#48;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"86\" style=\"vertical-align: -3px;\" \/>, and the temperature change is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-29665064f23dcd234606e3a3a35e1c17_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#56;&#51;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"99\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"import-auto-id2670063\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id1416811\">The hot resistance <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 obtained by entering known values into the above equation:<\/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-a58d72107c9bd6f0b6e04563bc3da289_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#82;&#38;&#32;&#61;&#38;&#32;&#123;&#82;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#48;&#32;&Omega;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#91;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#46;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#56;&#51;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#114;&#105;&#103;&#104;&#116;&#93;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#46;&#56;&#32;&Omega;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"59\" width=\"327\" style=\"vertical-align: -23px;\" \/><\/div>\n<p id=\"import-auto-id3054482\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2625691\">This value is consistent with the headlight resistance example in <a href=\"\/contents\/c8b9a2f8-cb8f-484f-b53c-72459218c189@4\">Ohm\u2019s Law: Resistance and Simple Circuits<\/a>.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Resistance in a Wire<\/div>\n<p>Learn about the physics of resistance in a wire. Change its resistivity, length, and area to see how they affect the wire&#8217;s resistance. The sizes of the symbols in the equation change along with the diagram of a wire.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id2470478\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/c55ee679dd7c4862e0e67fdc8fa3b8eb275eea29\/resistance-in-a-wire_en.jar\">Resistance in a Wire<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_21.3\" data-alt=\"\"><a href=\"\/resources\/c55ee679dd7c4862e0e67fdc8fa3b8eb275eea29\/resistance-in-a-wire_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/jpg\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/jpg\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id2423329\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id3176599\">\n<li id=\"import-auto-id3017858\">The resistance <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;\" \/> of a cylinder of length <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-66a9f474fc3c52efdfb0ba6a70199ee8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"12\" style=\"vertical-align: 0px;\" \/> and cross-sectional area <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25b206f25506e6d6f46be832f7119ffa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdb8da839bea2886124a1aa958d7444f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#92;&#114;&#104;&#111;&#32;&#76;&#125;&#125;&#123;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"55\" style=\"vertical-align: -6px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> is the resistivity of the material.<\/li>\n<li id=\"import-auto-id2684104\">Values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/> in <a href=\"#import-auto-id1375921\" class=\"autogenerated-content\">(Figure)<\/a> show that materials fall into three groups\u2014<em data-effect=\"italics\">conductors, semiconductors, and insulators<\/em>.<\/li>\n<li id=\"import-auto-id3008174\">Temperature affects resistivity; for relatively small temperature changes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cb5a269f1ca84874583e8696a1b59bff_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: 0px;\" \/>, resistivity is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3585d03aacf53b997ff1e5284e7772ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;&#61;&#123;&#92;&#114;&#104;&#111;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0fed00b387c11493e556843dd3ce1ab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#114;&#104;&#111;&#32;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: -4px;\" \/> is the original resistivity and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3b1d74aace57a3fb0bdaaf62680d0010_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&alpha;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/>  is the temperature coefficient of resistivity.<\/li>\n<li><a href=\"#import-auto-id1382426\" class=\"autogenerated-content\">(Figure)<\/a> gives values for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, the temperature coefficient of resistivity.<\/li>\n<li id=\"import-auto-id1609529\">The resistance <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;\" \/> of an object also varies with temperature: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-356822f8c20092ce207f835adb4a95f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#123;&#82;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-522741a1a3befc8cbda2a31eded25676_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#82;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> is the original resistance, and <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 the resistance after the temperature change.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1994296\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2446582\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2680317\">\n<p id=\"import-auto-id3224512\">In which of the three semiconducting materials listed in <a href=\"#import-auto-id1375921\" class=\"autogenerated-content\">(Figure)<\/a> do impurities supply free charges? (Hint: Examine the range of resistivity for each and determine whether the pure semiconductor has the higher or lower conductivity.)<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1575560\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3051109\">\n<p id=\"import-auto-id2684990\">Does the resistance of an object depend on the path current takes through it? Consider, for example, a rectangular bar\u2014is its resistance the same along its length as across its width? (See <a href=\"#import-auto-id1435027\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1435027\">\n<div class=\"bc-figcaption figcaption\">Does current taking two different paths through the same object encounter different resistance?<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1576396\" data-alt=\"Part a of the figure shows a voltage V applied along the length of a rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the length of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the length is shown as R and the current is shown as I. Part b of the figure shows a voltage V applied along the width of the same rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the width of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the width is shown as R prime, and the current is shown as I prime.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_21_03_04a.jpg\" data-media-type=\"image\/jpg\" alt=\"Part a of the figure shows a voltage V applied along the length of a rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the length of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the length is shown as R and the current is shown as I. Part b of the figure shows a voltage V applied along the width of the same rectangular bar using a battery. The current is shown to emerge from the positive terminal, pass along the width of the rectangular bar, and enter the negative terminal of the battery. The resistance of the rectangular bar along the width is shown as R prime, and the current is shown as I prime.\" width=\"325\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2452298\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1120880\">\n<p id=\"import-auto-id3105586\">If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1404656\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3201532\">\n<p id=\"import-auto-id1047618\">Explain why <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-356822f8c20092ce207f835adb4a95f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#123;&#82;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"144\" style=\"vertical-align: -4px;\" \/> for the temperature variation of the resistance <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;\" \/> of an object is not as accurate as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3585d03aacf53b997ff1e5284e7772ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;&#61;&#123;&#92;&#114;&#104;&#111;&#32;&#125;&#95;&#123;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#125;&#43;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#68;&#101;&#108;&#116;&#97;&#32;&#84;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"135\" style=\"vertical-align: -4px;\" \/>, which gives the temperature variation of resistivity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-43bc8be6acd1d7d6e61afc86bb1767f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#114;&#104;&#111;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id1815817\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2442974\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2930887\">\n<p id=\"import-auto-id1441418\">What is the resistance of a 20.0-m-long piece of 12-gauge copper wire having a 2.053-mm diameter?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1221864\">\n<p id=\"import-auto-id3423212\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-442acfd4eb71e6a651bcdd7fed0b2dd3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#49;&#48;&#52;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1871839\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2639671\">\n<p id=\"import-auto-id1525157\">The diameter of 0-gauge copper wire is 8.252 mm. Find the resistance of a 1.00-km length of such wire used for power transmission.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2668793\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1932262\">\n<p id=\"import-auto-id3017926\">If the 0.100-mm diameter tungsten filament in a light bulb is to have a resistance of<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-23f712ff50d7e61ec29cdf378ec94946_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#46;&#50;&#48;&#48;&#32;&Omega;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"41\" style=\"vertical-align: 0px;\" \/><br \/>\nat<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c8ebcea95df79fbc55f5432951696a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>, how long should it be?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1429348\">\n<p id=\"import-auto-id3424736\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0887c143b7f2229aa1c035e87aec9575_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#50;&#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;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"78\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3286438\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2454151\">\n<p id=\"import-auto-id3250821\">Find the ratio of the diameter of aluminum to copper wire, if they have the same resistance per unit length (as they might in household wiring).<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2396605\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3358944\">\n<p id=\"import-auto-id360022\">What current flows through a 2.54-cm-diameter rod of pure silicon that is 20.0 cm long, when<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d04d68fef0700046b46633fb22acd126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#49;&#46;&#48;&#48;&#32;&times;&#32;&#49;&#48;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#125;&#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;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"73\" style=\"vertical-align: -1px;\" \/><br \/>\n is applied to it? (Such a rod may be used to make nuclear-particle detectors, for example.)<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3034098\">\n<p id=\"import-auto-id3173375\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-64dd32dd97eef323b657c51129d34002_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;&#49;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#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;&#65;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"84\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3402655\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2599616\">\n<p id=\"import-auto-id934566\">(a) To what temperature must you raise a copper wire, originally at<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ab5d8026ce8291cd12b58cb16b7b20d8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#46;&#48;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>,<br \/>\nto double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances?<\/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-id2691474\">\n<p id=\"import-auto-id1447169\">A resistor made of Nichrome wire is used in an application where its resistance cannot change more than 1.00% from its value at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c8ebcea95df79fbc55f5432951696a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>. Over what temperature range can it be used?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2514795\">\n<p id=\"import-auto-id1934170\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cf9a3b0d0e17bde5b81dc7ca949b7b3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#32;&#116;&#111;&#32;&#52;&#53;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"92\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3449589\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2589289\">\n<p id=\"import-auto-id1220869\">Of what material is a resistor made if its resistance is 40.0% greater at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95453086a14fb8fcafc6b2bcfedcb280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"38\" style=\"vertical-align: -1px;\" \/> than at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c8ebcea95df79fbc55f5432951696a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3201853\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3085520\">\n<p id=\"import-auto-id1374451\">An electronic device designed to operate at any temperature in the range from <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-464972f61a2f871009c03d4f43698e33_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#32;&#116;&#111;&#32;&#53;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"121\" style=\"vertical-align: -1px;\" \/> contains pure carbon resistors. By what factor does their resistance increase over this range?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1917734\">\n<p id=\"import-auto-id1868093\">1.03<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1973609\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3385545\">\n<p id=\"import-auto-id1526233\">(a) Of what material is a wire made, if it is 25.0 m long with a 0.100 mm diameter and has a resistance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-25706a690f604123d2b243c8429c8bd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#55;&#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=\"48\" style=\"vertical-align: 0px;\" \/> at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c8ebcea95df79fbc55f5432951696a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>? (b) What is its resistance at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-20b7ca7945b4a508421b8c678662953a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"38\" style=\"vertical-align: -1px;\" \/>?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3103900\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3454882\">\n<p id=\"import-auto-id3177308\">Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c8ebcea95df79fbc55f5432951696a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3016931\">\n<p id=\"import-auto-id2956468\">0.06%<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3110369\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3044201\">\n<p id=\"import-auto-id3094331\">A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2448773\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2653584\">\n<p id=\"import-auto-id2454057\">A copper wire has a resistance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa382c7f6adaa4a8bf2e0982ec6dd63e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#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;&#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=\"57\" style=\"vertical-align: 0px;\" \/> at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3c8ebcea95df79fbc55f5432951696a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>, and an iron wire has a resistance of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f0ca4ad35b9f3704e8b6aeb62bf39d8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#50;&#53;&#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=\"57\" style=\"vertical-align: 0px;\" \/> at the same temperature. At what temperature are their resistances equal?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1998215\">\n<p id=\"import-auto-id2615498\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5e6ad05214e2864addb75c96b9073260_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#55;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"43\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2382175\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3201469\">\n<p id=\"import-auto-id1560690\">(a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-20b61ddcbd4ff39d46a5b7a2cc4026e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#45;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#54;&#48;&#48;&#125;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"119\" style=\"vertical-align: -5px;\" \/>) when it is at the same temperature as the patient. What is a patient\u2019s temperature if the thermistor\u2019s resistance at that temperature is 82.0% of its value at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c7c540827c3c0102e6028205e2899641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> (normal body temperature)? (b) The negative value<br \/>\nfor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-946f8144d4e3d460c8621773145884d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><em data-effect=\"italics\"> may not be maintained for very low temperatures. Discuss why and whether this is the case here. (Hint: Resistance can\u2019t become negative.)<\/em><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3189568\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1419790\">\n<p id=\"import-auto-id3388738\"><strong>Integrated Concepts<\/strong><\/p>\n<p id=\"eip-id2581512\">(a) Redo <a href=\"#fs-id1871839\" class=\"autogenerated-content\">(Figure)<\/a> taking into account the thermal expansion of the tungsten filament. You may assume a thermal expansion coefficient of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bb7336d56535cf71f604e0bc5f255004_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#54;&#125;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"74\" style=\"vertical-align: -5px;\" \/>. (b) By what percentage does your answer differ from that in the example?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1599741\">\n<p id=\"import-auto-id2963593\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4dfc8eae06f84ab0c58ab6b378da7768_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#55;&#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=\"14\" width=\"39\" style=\"vertical-align: -1px;\" \/> (total)<\/p>\n<p id=\"import-auto-id2930991\">(b) 3.0% decrease<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2415615\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1842428\">\n<p id=\"import-auto-id3090921\"><strong>Unreasonable Results<\/strong><\/p>\n<p id=\"eip-id1546986\">(a) To what temperature must you raise a resistor made of constantan to double its resistance, assuming a constant temperature coefficient of resistivity? (b) To cut it in half? (c) What is unreasonable about these results? (d) Which assumptions are unreasonable, or which premises are inconsistent?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"footnote-refs\">\n<h2 data-type=\"footnote-title\">Footnotes<\/h2>\n<ol>\n<li><a data-type=\"footnote-ref\" href=\"#footnote-ref1\">1<\/a> Values depend strongly on amounts and types of impurities<\/li>\n<li><a data-type=\"footnote-ref\" href=\"#footnote-ref2\">2<\/a> Values at  20\u00b0C.<\/li>\n<\/ol>\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-id1012671\">\n<dt>resistivity<\/dt>\n<dd id=\"fs-id3082005\">an intrinsic property of a material, independent of its shape or size, directly proportional to the resistance, denoted by <em data-effect=\"italics\">\u03c1<\/em><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id3305400\">\n<dt>temperature coefficient of resistivity<\/dt>\n<dd id=\"fs-id2448387\">an empirical quantity, denoted by <em data-effect=\"italics\">\u03b1<\/em>, which describes the change in resistance or resistivity of a material with temperature<\/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-1100","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1078,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1100","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\/1100\/revisions"}],"predecessor-version":[{"id":1101,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1100\/revisions\/1101"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1078"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1100\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1100"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1100"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1100"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1100"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}