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Electric Resistance

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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

quantity: resistance R
unit: ohm [Ω] Georg Ohm (1787–1854) Germany

Fancy version (the magnetohydrodynamic version?)…

J ∝ E

J = σE  ⇐ 
ρ = 1
σ
 ⇒  E = ρ J

Welcome to symbol hell…

Electrical properties
quantity symbol SI unit symbol property of…
resistance R ohm Ω objects
conductance G siemens S
resistivity ρ ohm meter Ωm materials
conductivity σ siemens per meter S/m

Ohm's law isn't a 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.

resistors

Bad booze rots our young guts but vodka goes well.

Better build roof over your garage before van gets wet.

Marking codes for resistors and capacitors
color digit multiplier tolerance tcr (10−6/K)
none ±20%
pink   10−3
silver   10−2 ±10%
gold   10−1 ±5%
black   0 100+ ±250
brown   1 101+ ±1% ±100
red   2 102+ ±2% ±50
orange   3 103+ ±0.05% ±15
yellow   4 104+ ±0.02% ±25
green   5 105+ ±0.50% ±20
blue   6 106+ ±0.25% ±10
violet   7 ±0.10% ±5
gray   8 ±0.01% ±1
white   9

materials

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 =  ne2
mevrms

where…

σ =  electrical conductivity [S/m]
n =  density of free electrons [e/m3]
e =  charge of an electron (1.60 × 10−19 C)
me =  mass of an electron (9.11 × 10−31 kg)
vrms =  root-mean-square speed of electrons [m/s]
ℓ =  mean free path length [m]

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)
aluminum 26.5
brass 64
chromium 126
copper 17.1
gold 22.1
iron 96.1
lead 208
lithium 92.8
mercury (0 °C) 941
manganese 1440
nichrome 1500
nickel 69.3
palladium 105.4
platinum 105
plutonium 1414
silver 15.9
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
nonmetals ρ (Ω m)
aluminum oxide (14 °C) 1 × 1014
aluminum oxide (300 °C) 3 × 1011
aluminum oxide (800 °C) 4 × 106
carbon, amorphous 0.35
carbon, diamond 2.7
carbon, graphite 650 × 10−9
indium tin oxide, thin film 2000 × 10−9
germanium 0.46
pyrex 7740 40,000
quartz 75 × 1016
silicon 640
silicon dioxide (20 °C) 1 × 1013
silicon dioxide (600 °C) 70,000
silicon dioxide (1300 °C) 0.004
water, liquid (0 °C) 861,900
water, liquid (25 °C) 181,800
water, liquid (100 °C) 12,740
Source: Encyclopedia of Agrophysics
foods ρ (Ω m)
apple17.9–26.3
beer5.56–7.7
bread crumb~57
butter~12.5
cucumber43.5
fruit juices 2.5–5.0
milk, fresh1.67–2.75
milk, sour 1.25–1.60
pear 37.0–71.4
potato 26.3–27.0
root vegetables24.4–66.7
syrups 16.7–25.0
tomato35.7
wheat, 10% moisture ~108
wheat, 24% moisture ~104

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 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))

Scatterplot showing the resistivity of copper at various temperatures with a line of best fit added

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

Scatterplot showing the resistivity of tungsten at various temperatures with a power curve fit added

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

miscellaneous

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