204. Resistance and ResistivityLearning Objectives
Material and Shape Dependence of ResistanceThe resistance of an object depends on its shape and the material of which it is composed. The cylindrical resistor in Figure 1 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’s electric resistance $R$ is directly proportional to its length $L$, 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, $R$ is inversely proportional to the cylinder’s crosssectional area $A$. 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 Table 1 gives representative values of $\rho $. 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. Table 1: Resistivities $\rho $ of Various materials at $\text{20\xba}\text{C}$
1. Values depend strongly on amounts and types of impurities Example 1: Calculating Resistor Diameter: A Headlight FilamentA car headlight filament is made of tungsten and has a cold resistance of $0\text{.}\text{350}\phantom{\rule{0.25em}{0ex}}\Omega $. If the filament is a cylinder 4.00 cm long (it may be coiled to save space), what is its diameter? Strategy We can rearrange the equation $R=\frac{\mathrm{\rho L}}{A}$ to find the crosssectional area $A$ of the filament from the given information. Then its diameter can be found by assuming it has a circular crosssection. Solution The crosssectional area, found by rearranging the expression for the resistance of a cylinder given in $R=\frac{\mathrm{\rho L}}{A}$, is $$A=\frac{\mathrm{\rho L}}{R}\text{.}$$Substituting the given values, and taking $\rho $ from Table 1, yields $$\begin{array}{lll}A& =& \frac{(5.6\times {\text{10}}^{\mathrm{\u20138}}\phantom{\rule{0.25em}{0ex}}\Omega \cdot \text{m})(4.00\times {\text{10}}^{\mathrm{\u20132}}\phantom{\rule{0.25em}{0ex}}\text{m})}{\text{1.350}\phantom{\rule{0.25em}{0ex}}\Omega}\\ & =& \text{6.40}\times {\text{10}}^{\mathrm{\u20139}}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}\text{.}\end{array}$$The area of a circle is related to its diameter $D$ by $$A=\frac{{\mathrm{\pi D}}^{2}}{4}\text{.}$$Solving for the diameter $D$, and substituting the value found for $A$, gives $$\begin{array}{lll}D& =& \text{2}{\left(\frac{A}{p}\right)}^{\frac{1}{2}}=\text{2}{\left(\frac{6.40\times {\text{10}}^{\mathrm{\u20139}}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}}{3.14}\right)}^{\frac{1}{2}}\\ & =& 9.0\times {\text{10}}^{\mathrm{\u20135}}\phantom{\rule{0.25em}{0ex}}\text{m}\text{.}\end{array}$$Discussion The diameter is just under a tenth of a millimeter. It is quoted to only two digits, because $\rho $ is known to only two digits. Temperature Variation of ResistanceThe resistivity of all materials depends on temperature. Some even become superconductors (zero resistivity) at very low temperatures. (See Figure 2.) 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 $\text{100\xba}\text{C}$ or less), resistivity $\rho $ varies with temperature change $\mathrm{\Delta}T$ as expressed in the following equation $$\rho ={\rho}_{0}(\text{1}+\alpha \mathrm{\Delta}T)\text{,}$$where ${\rho}_{0}$ is the original resistivity and $\alpha $ is the Table 2: Tempature Coefficients of Resistivity $\alpha $
2. Values at 20°C. Note also that $\alpha $ is negative for the semiconductors listed in Table 2, 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 $\rho $ with temperature is also related to the type and amount of impurities present in the semiconductors. The resistance of an object also depends on temperature, since ${R}_{0}$ is directly proportional to $\rho $. For a cylinder we know $R=\mathrm{\rho L}/A$, and so, if $L$ and $A$ do not change greatly with temperature, $R$ will have the same temperature dependence as $\rho $. (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 $L$ and $A$ is about two orders of magnitude less than on $\rho $.) Thus, $$R={R}_{0}(\text{1}+\alpha \mathrm{\Delta}T)$$is the temperature dependence of the resistance of an object, where ${R}_{0}$ is the original resistance and $R$ is the resistance after a temperature change $\mathrm{\Delta}T$. Numerous thermometers are based on the effect of temperature on resistance. (See Figure 3.) 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. Example 2: Calculating Resistance: HotFilament ResistanceAlthough caution must be used in applying $\rho ={\rho}_{0}(\text{1}+\alpha \mathrm{\Delta}T)$ and $R={R}_{0}(\text{1}+\alpha \mathrm{\Delta}T)$ for temperature changes greater than $\text{100\xba}\text{C}$, 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 ($\text{20\xbaC}$) to a typical operating temperature of $\text{2850\xba}\text{C}$? Strategy This is a straightforward application of $R={R}_{0}(\text{1}+\alpha \mathrm{\Delta}T)$, since the original resistance of the filament was given to be ${R}_{0}=0\text{.}\text{350 \Omega}$, and the temperature change is $\mathrm{\Delta}T=\text{2830\xba}\text{C}$. Solution The hot resistance $R$ is obtained by entering known values into the above equation: $$\begin{array}{lll}R& =& {R}_{0}(1+\alpha \mathrm{\Delta}T)\\ & =& (0\text{.}\text{350 \Omega})[\text{1}+(4.5\times {\text{10}}^{\mathrm{\u20133}}/\text{\xbaC})(\text{2830\xba}\text{C})]\\ & =& \text{4.8 \Omega .}\end{array}$$Discussion This value is consistent with the headlight resistance example in Ohm’s Law: Resistance and Simple Circuits. PhET Explorations: Resistance in a WireLearn 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. Section Summary
Conceptual QuestionsExercise 1In which of the three semiconducting materials listed in Table 1 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.) Exercise 2Does the resistance of an object depend on the path current takes through it? Consider, for example, a rectangular bar—is its resistance the same along its length as across its width? (See Figure 5.) Exercise 3If aluminum and copper wires of the same length have the same resistance, which has the larger diameter? Why? Exercise 4Explain why $R={R}_{0}(\text{1}+\alpha \mathrm{\Delta}T)$ for the temperature variation of the resistance $R$ of an object is not as accurate as $\rho ={\rho}_{0}(\text{1}+\alpha \mathrm{\Delta}T)$, which gives the temperature variation of resistivity $\rho $. Problems & ExercisesExercise 1What is the resistance of a 20.0mlong piece of 12gauge copper wire having a 2.053mm diameter? Show/Hide Solution Solution$\text{0.104 \Omega}$ Exercise 2The diameter of 0gauge copper wire is 8.252 mm. Find the resistance of a 1.00km length of such wire used for power transmission. Exercise 3If the 0.100mm diameter tungsten filament in a light bulb is to have a resistance of $\text{0.200 \Omega}$ at $\text{20}\text{.}\mathrm{0\xba}\text{C}$, how long should it be? Show/Hide Solution Solution$2\text{.}\text{8}\times {\text{10}}^{2}\phantom{\rule{0.25em}{0ex}}\text{m}$ Exercise 4Find 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). Exercise 5What current flows through a 2.54cmdiameter rod of pure silicon that is 20.0 cm long, when ${\mathrm{1.00\; \times \; 10}}^{\text{3}}\phantom{\rule{0.25em}{0ex}}\text{V}$ is applied to it? (Such a rod may be used to make nuclearparticle detectors, for example.) Show/Hide Solution Solution$1\text{.}\text{10}\times {\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{A}$ Exercise 6(a) To what temperature must you raise a copper wire, originally at $\text{20.0\xbaC}$, to double its resistance, neglecting any changes in dimensions? (b) Does this happen in household wiring under ordinary circumstances? Exercise 7A resistor made of Nichrome wire is used in an application where its resistance cannot change more than 1.00% from its value at $\text{20}\text{.}\mathrm{0\xba}\text{C}$. Over what temperature range can it be used? Show/Hide Solution Solution$\mathrm{5\xba}\text{C to 45\xbaC}$ Exercise 8Of what material is a resistor made if its resistance is 40.0% greater at $\text{100\xba}\text{C}$ than at $\text{20}\text{.}\mathrm{0\xba}\text{C}$? Exercise 9An electronic device designed to operate at any temperature in the range from $\text{\u201310}\text{.}\mathrm{0\xba}\text{C to 55}\text{.}\mathrm{0\xba}\text{C}$ contains pure carbon resistors. By what factor does their resistance increase over this range? Show/Hide Solution Solution1.03 Exercise 10(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 $\text{77}\text{.}7\phantom{\rule{0.25em}{0ex}}\Omega $ at $\text{20}\text{.}\mathrm{0\xba}\text{C}$? (b) What is its resistance at $\text{150\xba}\text{C}$? Exercise 11Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at $\text{20}\text{.}\mathrm{0\xba}\text{C}$? Show/Hide Solution Solution0.06% Exercise 12A wire is drawn through a die, stretching it to four times its original length. By what factor does its resistance increase? Exercise 13A copper wire has a resistance of $0\text{.}\text{500}\phantom{\rule{0.25em}{0ex}}\Omega $ at $\text{20}\text{.}\mathrm{0\xba}\text{C}$, and an iron wire has a resistance of $0\text{.}\text{525}\phantom{\rule{0.25em}{0ex}}\Omega $ at the same temperature. At what temperature are their resistances equal? Show/Hide Solution Solution$\text{17\xba}\text{C}$ Exercise 14(a) Digital medical thermometers determine temperature by measuring the resistance of a semiconductor device called a thermistor (which has $\alpha =\u20130\text{.}\text{0600}/\text{\xbaC}$) when it is at the same temperature as the patient. What is a patient’s temperature if the thermistor’s resistance at that temperature is 82.0% of its value at $\text{37}\text{.}\mathrm{0\xba}\text{C}$ (normal body temperature)? (b) The negative value for $\alpha $ may not be maintained for very low temperatures. Discuss why and whether this is the case here. (Hint: Resistance can’t become negative.) Exercise 15Integrated Concepts (a) Redo Exercise 2 taking into account the thermal expansion of the tungsten filament. You may assume a thermal expansion coefficient of $\text{12}\times {\text{10}}^{6}/\text{\xbaC}$. (b) By what percentage does your answer differ from that in the example? Show/Hide Solution Solution(a) $4\text{.}7\phantom{\rule{0.25em}{0ex}}\Omega $ (total) (b) 3.0% decrease Exercise 16Unreasonable Results (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?
