Effect of Temperature on the Resistance of Copper Wire

Transcription

1 PHYS-102 Honors Lab 1H Effect of Temperature on the Resistance of Copper Wire 1. Objective The objectives of this experiment are: To demonstrate the effect of temperature on the resistance of copper wire. To measure the temperature coefficient of resistance. 2. Theory (discussed in Sec of Serway and Jewett) For a limited range of temperatures, the resistivity of most pure metals varies linearly with temperature. This temperature dependence is described by the following expression: ρ = ρ o [1 + α (T T o )] (1) where ρ is the resistivity at temperature T (in degrees Celsius), ρ o is the resistivity at some reference temperature T o (usually 20ºC) and α is called the temperature coefficient of resistance, with units of (ºC) -1. For copper, the accepted value of α = (ºC) -1. The fact that resistivity increases with temperature can be understood in terms of a microscopic and classical model of current flow in metals. In this model, the structure of a metal can be thought of as a periodic network or lattice of atoms that vibrate with respect to their equilibrium positions. At a temperature of 0 K ( C) or so-called Absolute Zero, the motion of the electrons through the lattice is unimpeded. At higher temperatures, the lattice atoms vibrate about their equilibrium positions, resulting in the scattering of the electrons moving through the solid. Subject to an external electric potential difference or voltage V, accelerated electrons collide with and scatter off these vibrating lattice atoms. These collisions reduce the accelerated motion of the electrons to a net drift velocity which is manifested macroscopically as a current I. The ratio of the applied voltage V (measured in units of volts) to current I (measured in units of Ampere), is called the electrical resistance R, i.e. V / I = R (Ohm s Law) (2) The electrical resistance can also be expressed in terms of the geometry and intrinsic properties of a conducting wire. The resistance R is inversely proportional to the cross-sectional area of a cylindrical conductor and is directly proportional to both length L and the intrinsic resistivity ρ of the metal. These can be summarized in the expression: R = ρl/a (3) 1

2 Because resistance is proportional to resistivity, the resistivity ρ in Equation (1) can be replaced with resistance to yield: R = R o [ 1 + α (T T o )] (4) where R o is the resistance at the reference temperature (typically 20ºC). Rearranging this expression, we obtain: R/R o = αt + (1 - αt o ) (5) Thus, the temperature coefficient α corresponds to the slope of the normalized Resistance (R/Ro) vs. Temperature line and can be extracted from experimental data. 3. Experimental Details Overview of Experiment and Apparatus In this experiment, the resistance R of a coil of copper wire is measured at various temperatures ranging from room temperature down to the boiling point (-196ºC) of liquid nitrogen (LN). About 3.5 m of insulated copper wire with a diameter corresponding to gauge #40 and a room temperature resistance of about 13.9Ω is coiled around and thermally anchored one end of a brass rod using GE varnish (See Fig.1). The large mass of the brass rod is for thermal stability. The temperature of the wire is varied by lowering the brass rod gradually from near the mouth of a cylindrical glass dewar containing LN until the coil is submerged in LN. This takes advantage of the temperature gradient between the mouth of the dewar and the LN surface. Temperature is measured using a thermocouple mounted adjacent to the copper coil. Using a four-probe measurement technique, the resistance of the coil is measured and plotted against temperature and its linear dependence on temperature is verified. The temperature coefficient α is calculated from the slope of the R/Ro curve and compared to the standard value. To Voltmeter To Current Source in series w/ Ammeter To Thermocouple (thermometer) LN Fig. 1 Schematic of Experimental Setup. A 3.5m coil of copper wire is thermally anchored at the base of a brass rod. Four-probe measurement is used to measure the current through and voltage across it. 2

3 3.2 Two-Probe versus Four-Probe Measurement There are two general ways of measuring electrical resistance. The first is called Two-Probe Measurement and works well with samples with large resistances. Current is passed through the sample via two wire leads (current in and current out) and the voltage across the leads is measured. One then uses Ohm s Law R = V/I to determine the resistance R. The problem with this method is that the measured voltage includes the voltage drop across the leads. For a large sample resistance R and very small lead resistance, this method works well. However, if the resistance of the sample is small (e.g. 10Ω-100Ω), with slightly smaller or comparable lead resistances, the total voltage measured across the leads does not accurately reflect the voltage across the sample. To eliminate the voltage drop across the two leads, a Four-Probe measurement, which uses four wire leads instead of two, is typically performed. This is illustrated in Fig.2. A pair of wire leads A, B are connected to one end of the sample while another pair C, D are connected to the other. Current is passed from a contant-current source through A, through the sample and out through wire D. This current is measured by an ammeter in series with the current source. At the same time, the voltage across the sample, between wire B and C is measured using a voltmeter but because no current passes through the leads, the voltage drop measured is that of the sample itself. Because the sample resistance in this experiment is not large, the four-probe technique will be used. D C B A Fig. 2 Schematic for a Four-Probe Measurement of Resistance. This standard technique eliminates the voltage drop across the leads that measure voltage across the sample, in this case a coil of wire. 3.3 Measuring Temperature using a Thermocouple In this experiment, the temperature of the coil is measured using a type of thermometer called a thermocouple. It consists of two dissimilar metals, joined together at one end. When the junction of the two metals is heated or cooled a temperaturedependent potential difference or voltage develops across the junction. For this setup, a particular thermocouple called Type T is used and is in the form of a wire with copper and alloy copper-nickel alloy soldered at one end. For the Type T thermocouple, a 3

4 conversion table which tabulates the voltage, in millivolts, as a function of temperature is well known. This is reproduced in Table 1. Table 1. The Conversion Table of a Type T Thermocouple. The temperature-dependent voltage across the two dissimilar metals, measured in millivolts, is converted to the corresponding temperature, in units of ºC. Strictly speaking, one cannot just measure thermoelectric voltages because as one connects a voltmeter to the thermocouple, one creates new junctions. To solve this problem, thermocouples are often used with another junction sitting at a reference temperature (0ºC), realized using an ice-bath. The values tabulated in Table 1 were obtained using this recipe. However, to streamline this experiment, we will dispense with the ice bath and use room temperature as the reference temperature (typically 22 ºC). However, we must subtract from the actual voltage measured the corresponding voltage that Table 1 associates with room temperature (For Type T probe at 22ºC, this is millivolts). Using the conversion tables we then interpolate what temperature the processed voltage reading corresponds to. 3.4 Experimental Procedure 1. Using Fig. 1, carefully identify the different leads coming out of the copper wire coiled around the tip of the brass rod. Six leads should come out of the experimental package. Those labeled power or I+, I-need to be connected to the power supply supply circuit which contains a 15Ω protection resistor and an ammeter, as shown in Fig. 3. Those labeled V+, V- or voltage must be connected to a voltmeter. Those labeled Thermocouple must be connected to a high-resolution digital voltmeter. The power supply must be configured to operate in constant-current mode. The typical current to 4

5 be supplied to the coil should be around 50 ma. Before turning the power supply on and proceeding with the experiment, have your TA check your connections and settings. R p = 15Ω I+ I V+ V coil V- A I- Fig. 3 Schematic of the Four-Probe Measurement of the Coil Resistance. 2. At room temperature, read and record the current through the coil using the ammeter and the voltage across the coil, using the voltmeter. Record the reading on the highresolution voltmeter. 3. To proceed to lower temperatures, have your TA pour the appropriate amount of liquid nitrogen into the dewar. WARNING: It is extremely important that you do not touch liquid nitrogen directly and that whoever handles directly it must wear protective goggles to protect oneself from splashes. Because of its low temperature, LN will cause cryogenic burns. Do not play with it nor immerse your fingers in it. Listen carefully to the Safety Precautions outlined by your TA. 4. Carefully insert the brass rod into the dewar only to the extent of having the coil submerged in LN. Record your low temperature readings as in Step Record in the supplied Table in Part 4 Experimental Data, the coil voltage, coil resistance (obtained using Ohm s Law) and thermocouple voltage for 8-10 temperatures between room temperature and LN temperature. Do this by setting the brass rod progressively higher than the level of the LN surface. Another way is to wait till some of the LN evaporates and exposes the coil, thereby raising its temperature. 6. When all the data has been collected, turn off all power switches (voltmeters, power supply, etc.). KEEP EVERYTHING ABOVE THIS LINE. SUBMIT EVERYTHING BELOW THIS LINE. 5

6 Physics Honors Lab 1H Effect of Temperature on the Resistance of Copper Wire Name: Sec./Group Date: 4. Experimental Data Room Temperature Coil Resistance R o : Supply Current : ma Room Temperature : ºC Coil Voltage Resistance Themocouple (volts) (ohm) Voltage (mv) Ω Adjusted Voltage (mv) Temp (ºC) from Table R/R o Use Linear Regression to obtain: Slope of Best Fit line in R/R o vs T curve: Calculate R 2 value: 5. Analysis of Data 5.1 In the Data Table in Part 4, the Adjusted Voltage values (column 4) is calculated by adding to the corresponding Thermocouple voltage (column 3) the voltage corresponding to the Reference Temperature (22ºC or whatever is the room temperature), as read off the enclosed Type T Temperature Conversion Table. For example, if room temperature is 22ºC, then we add mv to the entry in Thermocouple voltage to obtain the Adjusted voltage. For Column 5, enter the corresponding temperature from the Conversion Table. For Column 6, divide the measured Resistance by the reference or room temperature resistance R o. 5.2 Plot using Excel or any other spreadsheet software the curve R/Ro versus T. Obtain, using linear regression analysis, the slope of the best fit line and the corresponding R 2 value. The slope corresponds to your experimental value for the temperature coefficient α. Submit the plot along with your report. 5.3 Calculate the percentage difference of your coefficient with respect to the accepted value for copper. 6

7 % difference = experimental value accepted value /accepted value x 100 % 5.4 What is the uncertainty in your measurement of the temperature coefficient? Make reasonable estimates of various uncertainties in your measurements of experimental quantities such as voltages and currents. Estimate the maximum uncertainty introduced by these values and compare this with your results. 7

8 Physics Honors Lab 1H - Prelab Effect of Temperature on the Resistance of Copper Wire Name: Sec./Group Date: 1. Liquid nitrogen boils at 77 ºK. What is this temperature in units of Celsius and Fahrenheit? 2. Suppose you were asked to measure the resistance of a 10 Ω resistor using a multimeter. The wires (leads) that connect the 10Ω resistor to your multimeter contribute an additional resistance of 1.5 Ω. a. Using this so-called two-probe measurement technique, what is the resulting percentage error? b. If the sample resistance was 1000Ω instead of 10Ω, what is the corresponding percentage error using the same leads? What does this tell you about when the two-probe measurement works well? 8

9 JUST FOR THE FUN OF IT: Cooling the Wires of a Giant Flashlight A. Using Copper Wires A demonstration electrical circuit consists of a flashlight bulb, battery and connecting wires which are all in series connection with a dense coil of copper wire. The circuit is mounted on a wooden stick with the copper wire coiled on one end. Turn the circuit on by screwing tightly the lightbulb onto its holder. Note the intensity of the lightbulb. Predict what happens to the light intensity of the bulb when you dip the end of the stick with the copper wire into liquid nitrogen. Then perform this demonstration to confirm your predictions. B. Using Manganin Wires For most pure metals such as copper, one can reduce the thermal vibrations of the lattice ions by dipping the wire into liquid nitrogen. However, if there are impurities in the metal such as in the case of an alloy, the component of the resistance due to to the impurities cannot be removed by lowering the temperature and keeping the regular lattice atoms from vibrating significantly. Predict what happens if the copper wire in this demonstration is replaced by manganin wire. Perform the same experiment except this time use the available manganin coil (the manganin wire length is shorter because it is more resistive at room temperature than copper wire). Verify your prediction. C. The Leidenfrost Effect Ask your TA to spill some liquid nitrogen on the floor. Watch as tiny beads of liquid nitrogen float on the floor, as if hovering on gas. This is called the Leidenfrost Effect. Watch as how it can be used to push away dust and light dirt along the floor. Explain how this happens. 9