**Resistance, Resistivity**

Circuits
electronics
resistors
capacitors

**Resistance of a wire in Ω**
R = ρL/A
ρ is resistivity of the material in Ω-m
L is length in meters
A is cross-sectional area in m²
A = πr², r is radius of wire in m
resistivity Cu 17.2e-9 Ω-m
conductivity = σ = 1/ρ
Conductance = 1/R = A/ρL = Aσ/L
**Stretched wire**
If you stretch it so that it is K longer
then cross-sectional area goes down by K
R goes up by K²
If you stretch it so that diameter reduces by K,
then area decreases by K², length goes up by K²
and R goes up by K⁴
**Sheet resistance**
Rs, in ohms per square, Ω/⃞) is the
resistance of a film of material.
Rs = resistivity / thickness
Resistance of the sheet is R = Rs(L/W)
**drift velocity Vd**
Vd = I / nqA
I is current
n is the number of charge carriers per m³
for copper, 8.5e28 electrons per m³
A is the cross sectional area in m²
A = πr², r is radius of wire in m
q is the charge of the charge carriers (electrons)
q = –1.602e–19 Coulomb (charge on an electron)
or
Vd = mσΔV / ρεfℓ
m is molecular mass of the conductor in kg
(mol mass in g / 1000)
σ is electric conductivity of the conductor in S/m (Siemens/meter)
(1/resistivity in Ω-m)
ΔV is the voltage across the conductor
ρ is the density in kg/m³
ε is the elementary charge, in C
(1.602e–19 Coulomb)
f is the number of free electrons per atom
ℓ is the length of the conductor, in m
**Power**
P = E*I
P = E²/R
P = I²R
1HP = 746 watts
1 amp = 1 coulomb per second
= 6.242e18 electrons per sec
**Ohm's law**
For constant R and varying current, E = IR, E and I
are proportional
For constant R and varying voltage, I = E/R, E and I
are proportional
For constant voltage and varying R, I = E/R, current
is inversely proportional to resistance.
For constant voltage and varying current, R = E/I,
resistance is inversely proportional to current.
For constant current and varying R, E = IR, voltage
is proportional to resistance
For constant current and varying voltage, R = E/I,
resistance is proportional to voltage
**Temperature Change in Resistance**
Rt = Ro(1 + α∆T)
α is temperature coef in /ºC
R = L/Aσ
σ = L/AR
σ is conductivity in 1/Ω-m
α for copper is 0.0039 /ºC
α for Aluminum is 0.0039 /ºC
α for nickel is 0.006 /ºC
α for iron is 0.005 /ºC
α for silver is 0.0038 /ºC
α for Tungston is 0.0045 /ºC
α for Zinc is 0.0037 /ºC
α for tin is 0.0045 /ºC
α for nichrome is 0.0004 /ºC
α for Carbon (amorphous) is –0.0005 /ºC
α for mercury is 0.0009 /ºC
α for Constantan is 0.000008 /ºC
α for Manganin is 0.000002 /ºC
**Volume Resistivity**
resistivity Ag 15.9e-9 Ω-m
resistivity Cu 17.2e-9 Ω-m or 17.2e-6 ohm-mm
resistivity Au 22.14e-9 Ω-m
resistivity Al 28.2e-9 Ω-m
resistivity brass 35e-9 Ω-m
resistivity W 56e-9 Ω-m
resistivity Zn 68e-9 Ω-m
resistivity Ni 69e-9 Ω-m
resistivity Fe 100e-9 Ω-m
resistivity Chromium 125e-9 Ω-m
resistivity Steel (cast) 160e-9 Ω-m
resistivity Pt 105e-9 Ω-m
resistivity Pb 208e-9 Ω-m
resistivity Hg 961e-9 Ω·m
resistivity Stainless steel (304) 750e-9 Ω-m
resistivity constantan 500e-9 Ω-m
resistivity Nichrome 1500e-9 Ω-m
resistivity carbon 6–40 µΩm
resistivity silicon 640–1000 Ωm
resistivity germanium 0.5-1 Ωm
Resistivity of pure water is 182 kΩ·m
Resistivity of sea water is 0.2 Ω·m
**density resistivity** (resistivity x density)
Al 28.2e-9 Ω-m * 2700 kg/m³ = 76e-6 Ω-kg/m²
Cu 17.2e-9 Ω-m * 8960 kg/m³ = 154e-6 Ω-kg/m²
Ag 15.9e-9 Ω-m * 10500 kg/m³ = 167e-6 Ω-kg/m²
Au 22.14e-9 Ω-m * 19300 kg/m³ = 427e-6 Ω-kg/m²
Zn 68e-9 Ω-m * 7130 kg/m³ = 485-6 Ω-kg/m²
Fe 100e-9 Ω-m * 7870 kg/m³ = 787e-6 Ω-kg/m²
W 56e-9 Ω-m * 19300 kg/m³ = 1080e-6 Ω-kg/m²
Pt 105e-9 Ω-m * 21500 kg/m³ = 2257-6 Ω-kg/m²
**Color Codes**
0 Black
1 Brown
2 Red
3 Orange
4 Yellow
5 Green
6 Blue
7 Violet
8 Gray
9 White
5% Gold x0.1
10% Silver x0.01
K (silver) 10%
J (gold) 5%
F (brown) 1%
G (red) 2%
D (green) 0.5%
C (blue) 0.25%
B (violet) 0.1%
(grey) 0.05%
Standard values 10%
10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82, 100
and multiples of these values
Standard values 5%
10, 11, 12, 13, 15, 16, 18, 20, 22, 24, 27, 30, 33,
36, 39, 43, 47, 51, 56, 62, 68, 75, 82, 91, 100