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## The Line of Resistance

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**The Line of Resistance**APS Teachers Day Workshop Los Angeles, CA March 22, 2005 Dr. Larry Woolf General Atomics Larry.Woolf@gat.com www.sci-ed-ga.org (click on Presentations to see all these slides)**Multimeter Operation**• Work with your group • With leads together, R = 0 • With leads not touching, R = open**Draw a line using the graphite pencil and measure its**resistance • Is the resistance measurement reproducible? Why or why not? • How could you optimize the line shape and the measurement technique to make the measurement more reproducible?**Design an experiment to determine how the resistance varies**with length • Discuss possible ways to do this with your group**Perform an experiment to determine how the resistance**varies with length • Discuss your data with your group • What model supports your data?**How does resistance vary with length?**• Write an equation that reflects this variation**Design an experiment to determine the total resistance of 2**resistors in series • Discuss possible ways to do this with your group**Perform an experiment to determine the total resistance of**2 resistors in series • Discuss your data with your group • What model supports your data?**What is the total resistance of 2 resistors in series?**• Write an equation that describes this relationship**Predict the resistance - if you double the length of a**resistorand - for 2 equal resistors in series**Single resistor R that doubles L: RT 2R 2 equal**resistors R in series: RT 2R**Design an experiment to determine how the resistance varies**with width Discuss possible ways to do this with your group**Perform an experiment to determine how the resistance**varies with width • Discuss your data with your group • What model supports your data?**How does resistance vary with width?**• Write an equation that reflects this variation**Design an experiment to determine the total resistance of 2**resistors in parallel • Discuss possible ways to do this with your group**Perform an experiment to determine the total resistance of**2 resistors in parallel • Discuss your data with your group • What model supports your data?**What is the total resistance of 2 resistors in parallel?**• Write an equation that describes this relationship • (Hint: Consider 1/R values of each resistor and of the resistors in parallel)**Predict the resistance - if you double the width of a**resistorand - for 2 equal resistors in parallel**Single resistor R that doubles W: RT R/2 2 equal**resistors R in parallel: RT R/2**How does resistance vary with length and width?**• Write an equation that reflects this variation**We found that R ~ L and R~ 1/Wso R ~ L/WHow does R vary with**thickness?Why do you think so?**Generally:**R = L/(Wt) = L/A (A=Wt) is called the electrical resistivity(t is thickness)**Resistivity and resistors-in-series relationshipR = L/AIf**L = L1 + L2R = (L1 + L2)/A = L1/A + L2/A = R1 + R2**Resistivity and resistors-in-parallel relationshipR =**L/AIf A = A1 + A2R = L/ (A1 + A2) 1/R = (A1 + A2)/ L1/R = A1/ L + A2/ L 1/R = 1/R1 + 1/R2**What is the difference between:**• Insulator • Semiconductor • Conductor**Creative drama for microscopic electron behavior for**insulator, semiconductor. and conductor**Conductor: ~1023 free electrons/cm3Semiconductor: ~ 1012 –**1022 free electrons/cm3Insulator: <1010 free electrons/cm3**Creative drama for microscopic electron behavior for width**dependence of resistance**Creative drama for microscopic electron behavior for length**dependence of resistance**Let’s look in more detail at the microscopic behavior of**electrons in a resistor**Electrical Resistance**• Resistance to flow of electrons when a voltage is applied • Apply a force (voltage) • Measure response to force (current) • Resistance is proportionality between force and response • Flow is due to: • Number of electrons that move past a point (plane) per second • (River current flow analogy – water current flow depends on width and depth of water, density of water, and the speed of the water: water flow is the number of water molecules that pass a point (plane perpendicular to motion) per second. In a similar manner, electron current flow depends on width and thickness of conductor, density of free electrons, and the speed of the electrons: electron flow is number of electric charges that pass a point (plane perpendicular to motion) per second.)**Known properties of circuits**V Resistor with resistance R I I L Measurements confirm constant I in the resistor. Therefore charges in wire move with constant velocity. But charges are subject to F=ma=qE=qV/L, so they should accelerate, not move with constant velocity! Why?**A model consistent with the data**Charges do not move freely from one end of the resistor to the other – they have lots of collisions, on average every time . Vfinal ~ a Therefore, charges move along the resistor with constant average “drift velocity - vD” that is proportional to the acceleration. (vD = a , not ½ a ; see references for details)**Electrical/Mechanical Analogy**V L L H Collision barriers**Pegboard model of Ohm’s LawAllows connection between:force**and motionandelectrical properties/Ohm’s Law**Pegboard Model of Electrical Resistance**• Balls – conduction electrons • Pegs – scattering centers in a solid • Height – voltage (V) • Pegboard length – resistor length (L) • Height/pegboard length – electric field (E=V/L) • Ideally, fixed density of balls – fixed density of conduction electrons in solid; then current is number of balls that pass a line (perpendicular to electric field) per unit time; and R=V/I**References for pegboard model**• Electricity and Magnetism, (Berkeley Physics Course volume 2), Edward M. Purcell, section 4.4: A Model for Electrical Conduction • “A mechanical analogy for Ohm’s Law,” M. do Couto Tavares et al., Phys. Educ. volume 26, 1991, p. 195-199. • http://www.iop.org/EJ/abstract/0031-9120/26/3/012 • “On an analogy for Ohm’s Law,” P. M. Castro de Oliveira, Phys. Educ. Volume 27, 1992, p. 60-61. • http://www.iop.org/EJ/abstract/0031-9120/27/2/001 • Feynman Lectures on Physics, volume 1, section 43, especially section 43-3. • Pegs: Vermont American ¼ inch x 1 ¼ inch wood peg • Available at Home Depot in the tool section: $2 for pack of 36 • Pegboard: 2 feet wide x 4 feet long • Available at Home Depot in lumber section: $6**Conclusion**• Simple experiments to examine length and width dependence of resistance and series and parallel combinations of resistors • Relationship between equation for resistivity and for series and parallel combinations of resistors • Pictorial (graphite lines) and mathematical connection • Microscopic behavior of electrons as the length and width of resistors are changed. • Creative dramas • Pegboard model: Connection between force and motion concepts and Ohm’s Law • This workshop is based on The Line of Resistance, available from the Institute of Chemical Education • http://ice.chem.wisc/edu/catalog.htm