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)
1
Multimeter Operation
• Work with your group
• With leads together, R = 0
• With leads not touching, R = open
2
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?
3
Design an experiment to
determine how the resistance
varies with length
• Discuss possible ways to do this with your
group
4
Perform an experiment to
determine how the resistance
varies with length
• Discuss your data with your group
• What model supports your data?
5
How does resistance vary with
length?
• Write an equation that reflects this variation
6
R~L
7
Design an experiment to
determine the total resistance of 2
resistors in series
• Discuss possible ways to do this with your
group
8
Perform an experiment to
determine the total resistance of
2 resistors in series
• Discuss your data with your group
• What model supports your data?
9
What is the total resistance of 2
resistors in series?
• Write an equation that describes this
relationship
10
RT = R1 + R2
11
Predict the resistance
- if you double the length of a resistor
and
- for 2 equal resistors in series
12
Single resistor R that doubles L: RT  2R
2 equal resistors R in series: RT  2R
13
Design an experiment to
determine how the resistance
varies with width
Discuss possible ways to do this with your
group
14
Perform an experiment to
determine how the resistance
varies with width
• Discuss your data with your group
• What model supports your data?
15
How does resistance vary with
width?
• Write an equation that reflects this variation
16
R ~ 1/W
or
1/R ~ W
17
Design an experiment to
determine the total resistance of 2
resistors in parallel
• Discuss possible ways to do this with your
group
18
Perform an experiment to
determine the total resistance of
2 resistors in parallel
• Discuss your data with your group
• What model supports your data?
19
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)
20
1/R1 + 1/R2 = 1/RT
21
Predict the resistance
- if you double the width of a resistor
and
- for 2 equal resistors in parallel
22
Single resistor R that doubles W: RT  R/2
2 equal resistors R in parallel: RT  R/2
23
How does resistance vary with
length and width?
• Write an equation that reflects this variation
24
We found that R ~ L and R~ 1/W
so R ~ L/W
How does R vary with thickness?
Why do you think so?
25
Generally:
R = L/(Wt) = L/A (A=Wt)
 is called the electrical resistivity
(t is thickness)
26
Resistivity and resistors-in-series relationship
R = L/A
If L = L1 + L2
R = (L1 + L2)/A
= L1/A + L2/A
= R1 + R2
27
Resistivity and resistors-in-parallel relationship
R = L/A
If A = A1 + A2
R = L/ (A1 + A2)
1/R = (A1 + A2)/ L
1/R = A1/ L + A2/ L
1/R = 1/R1 + 1/R2
28
Creative Dramas
29
What is the difference between:
• Insulator
• Semiconductor
• Conductor
30
Creative drama for microscopic
electron behavior for
insulator,
semiconductor.
and conductor
31
Conductor: ~1023 free electrons/cm3
Semiconductor: ~ 1012 – 1022 free electrons/cm3
Insulator: <1010 free electrons/cm3
32
When a resistor’s width is
doubled, what happens?
33
Creative drama for microscopic
electron behavior for
width dependence of resistance
34
When a resistor’s length is
doubled, what happens?
35
Creative drama for microscopic
electron behavior for
length dependence of resistance
36
Let’s look in more detail at the
microscopic behavior of electrons in
a resistor
37
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.)
38
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?
39
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)
40
Electrical/Mechanical Analogy
Electrical resistance
Mechanical ramp
Voltage V and resistor length L
Height H and ramp length L
F = qE = q V/L = ma
Framp = mgsin = mgH/L = ma
a ~ V/L
a ~ H/L
V
L
L
H
Collision barriers
41
Pegboard model of Ohm’s Law
Allows connection between:
force and motion
and
electrical properties/Ohm’s Law
42
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
43
Pegboard model of R=V/I
Change:
Effect:
L (at fixed V)
E so a  so vD  so I  so R
W (at fixed V)
I  so R 
V
E  so a  so vD  so I 
Density of balls
I  so R  (at fixed V)
(higher carrier density)
Density of pegs
vD  so I  so R (at fixed V)
(more defects)
or size of pegs 
(higher temperature – larger
vibration amplitude of ions)
44
Pegboard with Pegs
45
Close up of pegboard with pegs
46
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
47
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
48