RvsT-1 Week 3: Recitation Electric field of a continuous charge distribution Q1. Find the electric field at point (x 0, y 0 ) due a thin rod (length L) of charge with uniform linear charge density and total charge Q. Remember that the electric field is a vector. y P (x 0,y 0 ) - L/2!q L/2 x The following steps should help: (a) Find the components of the electric field at (x 0, y 0 ) due to the segment with length Δx and charge Δq that sits at position (x,0). (b) Express Δq in terms of the length of the segment Δx, total charge Q and length L of the rod. S: 10/2/12 12:20 PM 133 Temp and resistance.doc
RvsT-2 PHYSICS EXPERIMENTS 133 (c) You can find the total electric field by adding together the individual fields due to each of the segments that make up the rod. Do this by setting up appropriate integrals over the length of the rod. You don t need to evaluate the integrals. You will need to do this twice, once for each component of the electric field. (d) Evaluating the integrals will give you the following components of the electric field # E net,x = K Q % L % $ % # E net,y = K Q % L % $ % 1 ( x 0 " L 2) 2 2 [ + y 0 ] " 1/ 2 "( x 0 " L 2) [ ] 1/ 2 " y 0 ( x 0 " L 2) 2 2 + y 0 1 ( x 0 + L 2 ) 2 2 [ + y 0 ] 1/ 2 "( x 0 + L 2) & ( ( '( [ ] 1/ 2 y 0 ( x 0 + L 2) 2 2 + y 0 & ( ( '( Use these equations to evaluate the electric field at the point P = (0, y 0 ) on the perpendicular bisector, as shown in the figure above. Does the answer make sense?
RvsT-3 (e) Evaluate the electric field at P in the limit where y 0 is much much bigger than L. Does the result make sense? (Of course if you go very far away the electric field strength will go to zero. We want to see "how" it goes to zero.) (f) Evaluate the electric field at P in the limit where y 0 is much much smaller than L. Does the result make sense? Compare with the result for a line of charge.
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RvsT-5 Experiment 7 Temperature Dependence of Electrical Resistance GOAL. to determine the temperature dependence of the resistance of a metallic conductor. to determine the temperature dependence of the resistance of a semiconductor. to learn how to determine a material s resistance, and to apply this method to studying the temperature dependence of resistance for certain materials. EQUIPMENT. ring stand with insulated support water bath with heating element thermometer & stopper metallic (copper) conductor assembly (wire wrapped on a cylinder) semiconductor assembly (a disk held by stiff wires) power supply limiting resistor ammeter voltmeter 6 electrical leads THEORY. This experiment investigates how the resistance of a material changes with temperature. The temperature dependence of resistance is an important characteristic for circuit elements as it can effect the operation and performance of electrical devices. A material with a strong temperature dependent resistance can also be used for temperature measurement and control. The amount of current (I) flowing in a material is proportional to the potential difference ( V) between the ends of the material, i.e., I = "V R, (Eq. 1) where R is the resistance of the material. Equation 1 is known as Ohm s law. Thus, if we apply a potential difference ( V) across a material and measure the amount of current (I) flowing through the material we can determine the resistance of the material. This is shown schematically in Figure 1. L I A R!V Figure 1. Schematic of cylindrical material with resistance R, cross-sectional area A and length L. There is a current I flowing through the material and the potential difference between the ends of the material is ΔV.
RvsT-6 PHYSICS EXPERIMENTS --133 The resistance of a material depends on its length (L), cross sectional area (A) and a material property known as resistivity (ρ): R = "L A. (Eq. 2) The length (L) and cross sectional area (A) are geometric quantities, i.e. they depend on the size of the particular sample. Q1. Consider a cylindrical material with a fixed potential difference ( V) between its ends. If the length (L) is tripled and the diameter is halved, by what factor will the current (I) change? In this experiment the length (L) and the cross sectional area (A) of the material will be kept fixed. Thus, any changes in resistance will be due to changes in the resistivity (ρ), which depends on the inherent properties of charge flow in the material. For metals the resistivity (ρ) depends on the mass (m) of the electron, the magnitude of the electron charge (e), the number density of electrons (n) and the mean time (τ) it takes for an electron to collide with an atom as it moves through the metal. The relationship between the resistivity and these properties is: " = m ne 2 #. (Eq. 3) The mass (m) and charge (e) of the electron are fundamental constants and do not change with temperature. For a metal at moderate temperatures (such as those in this experiment) the number density of electrons (n) is effectively independent of temperature. Note that this is not true for semiconductors. The mean time (τ) between collisions for the electron does depend on temperature. As the temperature increases, the atoms in the metal vibrate more, which makes them bigger targets for the electrons. Q2. Do you expect the resistance of a metal to increase or decrease as temperature is increased? Use the information above and explain your answer in sentence form. From your answer to Q2 above you should see that is reasonable to expect a linear relationship between resistance (R) and temperature (T). Such a relationship is usually written as: R = R 20 [1+ "(T # 20 o C)], (Eq. 4) where R 20 is the value of resistance (R) at a reference temperature T = 20 o C. The quantity α is called the temperature coefficient of resistance. It is a material property that gives the strength of the dependence of resistance (R) on temperature (T). By measuring the resistance (R) at different values of temperature (T), the temperature coefficient of resistance (α) can be determined. Equation 4 can be rewritten as: "R "T = R 20#, (Eq. 5) where "R "T would be the slope of a graph of resistance (R) versus temperature (T).
RvsT-7 PART I. PROCEDURE. Fill the vessel with an ice water mixture to within 2 or 3 cm of the top. Support it in the fiber ring on top of the tripod. Insert the thermometer in the 1-hole stopper and place it in the black bakelite cover support of the metallic-conductor unit. Insert the unit in the vessel, clamping it in place. The heating vessel and metallic conductor unit (referred to as the sample ) are shown in Figure 2. thermometer stirring rod binding post ice/water sample copper wire semiconductor Figure 2. The sample holder with heating vessel and metallic conductor unit. Assemble the circuit using Figures 3 and/or 4. Figure 3 shows the circuit diagram while Figure 4 shows a "picture" of the wiring. Make sure that both the ammeter (A) and voltmeter (V) are set to DC. Start with the heating vessel unplugged. limiting resistor Ammeter A Power Supply Voltmeter V sample Figure 3. The circuit diagram for the experiment.
RvsT-8 PHYSICS EXPERIMENTS --133 Power Supply Limiting Resistor Ammeter Voltmeter Figure 4. Schematic of wiring for the experiment. Sample Q3. What would be the ideal internal resistance of the voltmeter? Why? Stir the water; when the water and apparatus have come to thermal equilibrium (no change in temperature) read and record the following data in the first empty columns of the first row of Table 1 below: the temperature (T) to the nearest 0.1 C, the current (I) through the sample of metallic conductor (using the ammeter), the potential difference (ΔV) across the sample (using the voltmeter). Calculate the resistance (R = ΔV/I) of the sample and enter it into the last column of the first row of Table 1. Plug in the heating vessel. (Note that the heating element is only on while the button is pushed down.) At intervals of 5-10 C record the temperature (T), the current (I), the potential difference (ΔV) in Table 1 below. In each instance be sure that the temperature is constant during these measurements. To do this it will be necessary to turn off the heater one or two degrees before the desired temperature is reached, and then stir until maximum temperature is obtained. Obtain measurements until a temperature (T) of about 80 C is reached. Graph the data with the resistance R (dependent variable) on the vertical axis and temperature (T) on the horizontal axis, on Graph 1 below. You may use the computer to generate a graph if you wish. Using the straight line through the data points obtain the value of R 20 (the resistance at T = 20 C). Using the slope of the resistance (R) versus temperature (T) graph and the value of R 20 (the resistance at T = 20 C), calculate the temperature coefficient of resistance (α). Calculate the percent difference between your value for temperature coefficient of resistance (α) and that of the accepted value, " Cu = 3.9 #10 $3 o C $1. PART II. It has been shown that for a semiconductor the resistance (R) decreases with increasing temperature (T). The relationship is: T 0 T R = R " e, (Eq. 6) where R " is the value that the resistance (R) approaches as the temperature (T) gets very large and depends on the sample, T 0 is a characteristic temperature for the semiconductor. The quantity T 0 can be different for each type of semiconductor material (silicon, germanim, etc.). The above relationship for the semiconductor only applies if the temperature is expressed in Kelvin. (T Kelvin = T Celsius +273) Task: Investigate the relationship between resistance (R) and temperature (T) for the semiconductor and also determine T 0.
RvsT-9 Q4. Using Eq. 6 show that a plot of ln(r) vs 1/T should give a straight line. Make sure to include your calculations in your report. What quantity does the slope of the line correspond to? Your report should include: A description of measurements and data displayed in lists and/or tables. An analysis containing graphs and calculations from graphs and/or equations. Results in mathematical and/or graphical form with comparison to the prediction in Equation 6 and/or Q4.
RvsT-10 PHYSICS EXPERIMENTS --133 NAME: REPORT. ANSWERS TO QUESTIONS (Q1-3). COURSE/SECTION:
RvsT-11 DATA. Temperatu re T ( 0 C) Current I (ma) Voltage V (mv) Resistance R (Ω) Table 2. Metal Data Table
R (Ω) T ( 0 C) RvsT-12 PHYSICS EXPERIMENTS --133 ANALYSIS. Sample calculation of resistance (R). Graph 1. Temperature Dependence of Resistance Determination of the temperature coefficient of resistance (α).
RvsT-13 ANSWER TO QUESTION Q4.