Electrical Resistance

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Discussion

introduction

Yech! What a mess this is.

Conduction: S. Gray, 1729 -- Resistance: Georg Simon Ohm, 1827.

Regular version …

I ∝ V

I =  V ⇒ V = IR ⇒ R =  V
R I

Symbology …

Fancy version (the magnetohydrodynamic theory version?) …

J ∝ E

J = σE = − σ ∇V & ρ =  1 ⇒ J =  E or E = ρJ
σ ρ

Symbol hell …

Ohm's law isn't a very serious law. It's the jaywalking of physics. Sensible materials and devices obey it, but there are plenty of rogues out there that don't.

Resistor Color Code
first & second bands
(first & second digits)		   third band
(multiplier)		   fourth band
(tolerance)
black   0   black   1   none   ±20%
brown   1   brown   10   silver   ±10%
red   2   red   100   gold   ±5%
orange   3   orange   1,000        
yellow   4   yellow   10,000        
green   5   green   100,000        
blue   6   blue   1,000,000        
violet   7   silver   0.01        
gray   8   gold   0.1        
white   9                

Bad Booze Rots Our Young Guts But Vodka Goes Well.

Better Build Roof Over Your Garage Before Van Gets Wet.

solids

Resistance and resistivity. Factors affecting resistance in a conducting wire.

R =  ρℓ
A

Conductors vs. insulators

Best electrical conductors: silver, copper, gold, aluminum, calcium, beryllium, tungsten

Resistivity and conductivity are reciprocals.

Conductivity in metals is a statistical/thermodynamic quantity.

Resistivity is determined by the scattering of electrons. The more scattering, the higher the resistance.

σ =  ne2ℓ
mevrms

Where …

σ  =  electrical conductivity
n  =  density of free electrons
e  =  charge of an electron
me  =  mass of an electron
vrms  =  root-mean-square speed of electrons
ℓ  =  mean free path length

Graphite

Where does this idea belong? Nichrome was invented in 1906, which made electric toasters possible.

Conducting polymers.

Resistivity of Selected Materials (~300 K)
(Note the difference in units between metals and nonmetals.)
metals ρ (nΩ·m)   nonmetals ρ (Ω·m)
aluminum 26.5   aluminum oxide (14 °C) 1 × 1014
brass 64   aluminum oxide (300 °C) 3 × 1011
chromium 126   aluminum oxide (800 °C) 4 × 106
copper 17.1   carbon, amorphous 0.35
gold 22.1   carbon, diamond 2.7
iron 96.1   carbon, graphite 650 × 10−9
lead 208   germanium 0.46
lithium 92.8   pyrex 7740 40,000
mercury (0 °C) 941   quartz 75 × 1016
manganese 1440   silicon 640
nichrome 1500   silicon dioxide (20 °C) 1 × 1013
nickel 69.3   silicon dioxide (600 °C) 70,000
palladium 105.4   silicon dioxide (1300 °C) 0.004
platinum 105   water, liquid (0 °C) 861,900
plutonium 1414   water, liquid (25 °C) 181,800
silver 15.9   water, liquid (100 °C) 12,740
solder 150      
steel, plain 180      
steel, stainless 720      
tantalum 131      
tin (0 °C) 115      
titanium (0 °C) 390      
tungsten 52.8      
uranium (0 °C) 280      
zinc 59      

temperature

The general rule is resistivity increases with increasing temperature in conductors and decreases with increasing temperature in insulators. Unfortunately there is no simple mathematical function to describe these relationships.

The temperature dependence of resistivity (or its reciprocal, conductivity) can only be truly understood with quantum mechanics. In the same way that matter is an assembly of microscopic particles called atoms and a beam of light is a stream of microscopic particles called photons, thermal vibrations in a solid are a swarm of microscopic particles called phonons. The electrons are trying to drift toward the positive terminal of the battery, but the phonons keep crashing into them. The random direction of these collisions disturbs the attempted organized motion of the electrons against the electric field. The deflection or scattering of electrons with phonons is one source of resistance. As temperature rises, the number of phonons increases and with it the likelihood that the electrons and phonons will collide. Thus when temperature goes up, resistance goes up.

For some materials, resistivity is a linear function of temperature.

ρ = ρ0(1 + α(T − T0))

The resistivity of a conductor increases with temperature. In the case of copper, the relationship between resistivity and temperature is approximately linear over a wide range of temperatures.

For other materials, a power relationship works better.

ρ = ρ0(T ∕ T0)μ

The resistivity of a conductor increases with temperature. In the case of tungsten, the relationship between resistivity and temperature is best described by a power relationship.

see also: superconductivity

magnetoresistance

photoconductivity

liquids

electrolytes

gases

dielectric breakdown

plasmas

microphones

A carbon microphone is a backward nothing

Microphones and How They Work
type sounds produce
changes in … 		which cause
changes in … 		which result in
changes in …
carbon granule density resistance voltage
condenser plate separation capacitance voltage
dynamic coil location flux voltage
piezoelectric compression polarization voltage

Summary

Problems

practice

  1. A standard 60 W 120 V light bulb has a tungsten filament that is 53.3 cm long and 46 μm in diameter.
    1. What is the light bulb's operating resistance?
    2. Determine the cross sectional area of the filament.
    3. Determine the resistivity of tungsten.
    4. How does the resistivity calculated above compare to the value quoted in standard reference tables? Why are these two values so different?
    5. How can a 53.3 cm filament fit into a light bulb that is only a few centimeters wide?
    Solutions …
    1. Answer it.
    2. Answer it.
    3. Answer it.
    4. Answer it.
    5. Answer it.
  2. Write something.
    • Answer it.
  3. Write something.
    • Answer it.
  4. Write something completely different.
    • Answer it.

conceptual

  1. Given a wire with a resistance R, what will be the new resistance if …
    1. the wire is cut in half and only one half is used to conduct electricity,
    2. the wire is folded in half and both halves are used to conduct electricity?
  2. A tungsten rod and an aluminum rod have the same length and resistance.
    1. What is the ratio of the cross sectional area of the tungsten rod to the aluminum rod?
    2. What is the ratio of the diameter of the tungsten rod to the aluminum rod?

numerical

  1. A power transmission cable is composed of 37 strands of aluminum wire, each 4.0 mm in diameter. The cable is 100 m long and is used to deliver 300 A of current to a commercial power user. Determine …
    1. the total cross sectional area of the cable,
    2. the resistance of the cable, and
    3. the power lost in the cable before it reaches the user.
  2. You have decided to build an 800 W 120 V, two slot toaster for your mother.
    1. How much 25 gauge (0.455 mm diameter) nichrome wire will you need?
    2. Approximately how many times should the wire be folded so that both sides of each slice of bread will be toasted evenly? (Assume that a typical slice of bread is a 12 cm square.)
  3. Which dry pasta offers more resistance to the flow of electricity?
    • spaghetti lunghi, an extra long variety of string-like pasta, 50 cm long, 1.55 mm in diameter
    • capelli d'angelo: a very fine hair-like pasta, 25 cm long, 0.965 mm in diameter
    Assume that both pastas are made from durum semolina wheat prepared under identical conditions.
  4. What is the resistance of a roll of 100 pennies? To simplify calculations, assume the pennies are made entirely of copper.
    • If you use American pennies, assume they were minted before 1982. Pennies minted after 1982 have a zinc core in a copper jacket.
    • If you use Canadian pennies, assume they were minted before 1996. Pennies minted after 1996 have a zinc or nickel-steel core in a copper jacket.
  5. An electric power distribution cable is made of multiple strands of aluminum and steel wire as shown in the diagram below.
     
    The diameters are 2.00 mm and 1.33 mm for the aluminum and steel strands, respectively. Determine the resistance for one kilometer of this cable …
    1. assuming that each strand is straight
    2. assuming that each aluminum strand is wound with a 16° pitch
  6. What dimensions should a 50 nm aluminum film have to yeld a resistance of 40 Ω?

statistical

  1. zero-the-meters.txt
    A group of students were assigned the task of testing various off the shelf resistors. They were told to gradually increase the voltage across the resistor and measure both the voltage and the current. Unfortunately, they wired the meters in backward and forgot to zero them before taking measurements. The situation is not all that bad, however. Using a spreadsheet program or other similar data analysis software …
    1. repair the voltage and current data to compensate for the students' errors,
    2. construct a graph from the repaired voltage and current data, and
    3. determine the resistance of the resistor to the nearest ohm.

Resources

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