1 The Line of Resistance APS Teachers Day Workshop Los Angeles, CA March 22, 2005 Dr. Larry Woolf General Atomics (click on Presentations to see all these slides)

2 Multimeter Operation Work with your group With leads together, R = 0 With leads not touching, R = open

3 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?

4 Design an experiment to determine how the resistance varies with length Discuss possible ways to do this with your group

5 Perform an experiment to determine how the resistance varies with length Discuss your data with your group What model supports your data?

6 How does resistance vary with length? Write an equation that reflects this variation

7 R ~ L

8 Design an experiment to determine the total resistance of 2 resistors in series Discuss possible ways to do this with your group

9 Perform an experiment to determine the total resistance of 2 resistors in series Discuss your data with your group What model supports your data?

10 What is the total resistance of 2 resistors in series? Write an equation that describes this relationship

11 R T = R 1 + R 2

12 Predict the resistance - if you double the length of a resistor and - for 2 equal resistors in series

13 Single resistor R that doubles L: R T  2R 2 equal resistors R in series: R T  2R

14 Design an experiment to determine how the resistance varies with width Discuss possible ways to do this with your group

15 Perform an experiment to determine how the resistance varies with width Discuss your data with your group What model supports your data?

16 How does resistance vary with width? Write an equation that reflects this variation

17 R ~ 1/W or 1/R ~ W

18 Design an experiment to determine the total resistance of 2 resistors in parallel Discuss possible ways to do this with your group

19 Perform an experiment to determine the total resistance of 2 resistors in parallel Discuss your data with your group What model supports your data?

20 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)

21 1/R 1 + 1/R 2 = 1/R T

22 Predict the resistance - if you double the width of a resistor and - for 2 equal resistors in parallel

23 Single resistor R that doubles W: R T  R/2 2 equal resistors R in parallel: R T  R/2

24 How does resistance vary with length and width? Write an equation that reflects this variation

25 We found that R ~ L and R~ 1/W so R ~ L/W How does R vary with thickness? Why do you think so?

26 Generally: R =  L/(Wt) =  L/A (A=Wt)  is called the electrical resistivity (t is thickness)

27 Resistivity and resistors-in-series relationship R =  L/A If L = L 1 + L 2 R =  (L 1 + L 2 )/A =  L 1 /A +  L 2 /A = R 1 + R 2

28 Resistivity and resistors-in-parallel relationship R =  L/A If A = A 1 + A 2 R =  L/ (A 1 + A 2 ) 1/R = (A 1 + A 2 )/  L 1/R = A 1 /  L + A 2 /  L 1/R = 1/R 1 + 1/R 2

29 Creative Dramas

30 What is the difference between: Insulator Semiconductor Conductor

31 Creative drama for microscopic electron behavior for insulator, semiconductor. and conductor

32 Conductor: ~10 23 free electrons/cm 3 Semiconductor: ~ – free electrons/cm 3 Insulator: <10 10 free electrons/cm 3

33 When a resistor’s width is doubled, what happens?

34 Creative drama for microscopic electron behavior for width dependence of resistance

35 When a resistor’s length is doubled, what happens?

36 Creative drama for microscopic electron behavior for length dependence of resistance

37 Let’s look in more detail at the microscopic behavior of electrons in a resistor

38 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.)

39 Known properties of circuits V II Resistor with resistance R 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?

40 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 . V final ~ a  Therefore, charges move along the resistor with constant average “drift velocity - v D ” that is proportional to the acceleration. (v D = a , not ½ a  ; see references for details)

41 H L Electrical resistance Voltage V and resistor length L F = qE = q V/L = ma a ~ V/L Electrical/Mechanical Analogy Mechanical ramp Height H and ramp length L F ramp = mgsin  = mgH/L = ma a ~ H/L V L Collision barriers

42 Pegboard model of Ohm’s Law Allows connection between: force and motion and electrical properties/Ohm’s Law

43 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

44 Pegboard model of R=V/I Change:Effect: L  (at fixed V)E  so a  so v D  so I  so R  W  (at fixed V)I  so R  V  E  so a  so v D  so I  Density of balls  (higher carrier density) I  so R  (at fixed V) Density of pegs  (more defects) or size of pegs  (higher temperature – larger vibration amplitude of ions) v D  so I  so R  (at fixed V)

45 Pegboard with Pegs

46 Close up of pegboard with pegs

47 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 – “On an analogy for Ohm’s Law,” P. M. Castro de Oliveira, Phys. Educ. Volume 27, 1992, p – Feynman Lectures on Physics, volume 1, section 43, especially section 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

48 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 –