Resistance opposes electric current, like friction for electricity. Higher resistance = harder for current to flow. Measured in ohms (Ω). Every material has resistance—even wires. Zero resistance only in superconductors.

• 1 ohm = 1 volt per ampere (1 Ω = 1 V/A)
• Resistance limits current (R = V/I)
• Conductors: low R (copper ~0.017 Ω·mm²/m)
• Insulators: high R (rubber >10¹³ Ω·m)

Conductance (G) = 1/Resistance. Measured in siemens (S). 1 S = 1/Ω. Two ways to describe same thing: high resistance = low conductance. Use whichever is convenient!

• Conductance G = 1/R (siemens)
• 1 S = 1 Ω⁻¹ (reciprocal)
• High R → low G (insulators)
• Low R → high G (conductors)

Resistance changes with temperature! Metals: R increases with heat (positive temp coefficient). Semiconductors: R decreases with heat (negative). Superconductors: R = 0 below critical temp.

• Metals: +0.3-0.6% per °C (copper +0.39%/°C)
• Semiconductors: decreases with temperature
• NTC thermistors: negative coefficient
• Superconductors: R = 0 below Tc

Ohm (Ω) is SI derived unit for resistance. Named after Georg Ohm (Ohm's law). Defined as V/A. Prefixes from femto to tera cover all practical ranges.

• 1 Ω = 1 V/A (exact definition)
• TΩ, GΩ for insulation resistance
• kΩ, MΩ for typical resistors
• mΩ, µΩ, nΩ for wires, contacts

Siemens (S) is reciprocal of ohm. 1 S = 1/Ω = 1 A/V. Named after Werner von Siemens. Formerly called 'mho' (ohm backwards). Useful for parallel circuits.

• 1 S = 1/Ω = 1 A/V
• Old name: mho (℧)
• kS for very low resistance
• mS, µS for moderate conductance

Abohm (EMU) and statohm (ESU) from old CGS system. Rarely used today. 1 abΩ = 10⁻⁹ Ω (tiny). 1 statΩ ≈ 8.99×10¹¹ Ω (huge). SI ohm is standard.

• 1 abohm = 10⁻⁹ Ω = 1 nΩ (EMU)
• 1 statohm ≈ 8.99×10¹¹ Ω (ESU)
• Obsolete; SI ohm is universal
• Only in old physics texts

V = I × R (voltage = current × resistance). Fundamental relationship. Know any two, find the third. Linear for resistors. Power dissipation P = I²R = V²/R.

• V = I × R (voltage from current)
• I = V / R (current from voltage)
• R = V / I (resistance from measurements)
• Power: P = I²R = V²/R (heat)

Series: R_total = R₁ + R₂ + R₃... (resistances add). Parallel: 1/R_total = 1/R₁ + 1/R₂... (reciprocals add). For parallel, use conductance: G_total = G₁ + G₂.

• Series: R_tot = R₁ + R₂ + R₃
• Parallel: 1/R_tot = 1/R₁ + 1/R₂
• Parallel conductance: G_tot = G₁ + G₂
• Two parallel equal R: R_tot = R/2

R = ρL/A (resistance = resistivity × length / area). Material property (ρ) + geometry. Long thin wires have high R. Short thick wires have low R. Copper: ρ = 1.7×10⁻⁸ Ω·m.

• R = ρ × L / A (geometry formula)
• ρ = resistivity (material property)
• L = length, A = cross-sectional area
• Copper ρ = 1.7×10⁻⁸ Ω·m

Resistors: 1 Ω to 10 MΩ typical. Pull-up/down: 10 kΩ common. Current limiting: 220-470 Ω for LEDs. Voltage dividers: kΩ range. Precision resistors: 0.01% tolerance.

• Standard resistors: 1 Ω - 10 MΩ
• Pull-up/pull-down: 1-100 kΩ
• LED current limiting: 220-470 Ω
• Precision: 0.01% tolerance available

Shunt resistors: mΩ range (current sensing). Wire resistance: µΩ to mΩ per meter. Contact resistance: µΩ to Ω. Cable impedance: 50-75 Ω (RF). Grounding: <1 Ω required.

• Current shunts: 0.1-100 mΩ
• Wire: 13 mΩ/m (22 AWG copper)
• Contact resistance: 10 µΩ - 1 Ω
• Coax: 50 Ω, 75 Ω standard

Superconductors: R = 0 exactly (below Tc). Insulators: TΩ (10¹² Ω) range. Human skin: 1 kΩ - 100 kΩ (dry). Electrostatic: GΩ measurements. Vacuum: infinite R (ideal insulator).

• Superconductors: R = 0 Ω (T < Tc)
• Insulators: GΩ to TΩ
• Human body: 1-100 kΩ (dry skin)
• Air gap: >10¹⁴ Ω (breakdown ~3 kV/mm)

Each prefix step = ×1000 or ÷1000. MΩ → kΩ: ×1000. kΩ → Ω: ×1000. Ω → mΩ: ×1000.

• MΩ → kΩ: multiply by 1,000
• kΩ → Ω: multiply by 1,000
• Ω → mΩ: multiply by 1,000
• Reverse: divide by 1,000

G = 1/R (conductance = 1/resistance). R = 1/G. 10 Ω = 0.1 S. 1 kΩ = 1 mS. 1 MΩ = 1 µS. Reciprocal relationship!

• G = 1/R (siemens = 1/ohms)
• 10 Ω = 0.1 S
• 1 kΩ = 1 mS
• 1 MΩ = 1 µS

R = V / I. Know voltage and current, find resistance. 5V at 20 mA = 250 Ω. 12V at 3 A = 4 Ω.

• R = V / I (Ohms = Volts ÷ Amps)
• 5V ÷ 0.02A = 250 Ω
• 12V ÷ 3A = 4 Ω
• Remember: divide voltage by current

5V supply, LED needs 20 mA and has 2V forward voltage. What resistor?

Voltage drop = 5V - 2V = 3V. R = V/I = 3V ÷ 0.02A = 150 Ω. Use standard 220 Ω (safer, less current).

Two 10 kΩ resistors in parallel. What's total resistance?

Equal parallel: R_tot = R/2 = 10kΩ/2 = 5 kΩ. Or: 1/R = 1/10k + 1/10k = 2/10k → R = 5 kΩ.

12V across 10 Ω resistor. How much power?

P = V²/R = (12V)² / 10Ω = 144/10 = 14.4 W. Use 15W+ resistor! Also: I = 12/10 = 1.2A.

The 'quantum of resistance' h/e² ≈ 25,812.807 Ω is a fundamental constant. At quantum scale, resistance comes in multiples of this value. Used in quantum Hall effect for precise resistance standards.

Below critical temperature (Tc), superconductors have R = 0 exactly. Current flows forever with no loss. Once started, a superconducting loop maintains current for years without power. Enables powerful magnets (MRI, particle accelerators).

Lightning channel resistance drops to ~1 Ω during strike. Air normally >10¹⁴ Ω, but ionized plasma is conductive. Channel heats to 30,000 K (5× sun surface). Resistance increases as plasma cools, creating multiple pulses.

At high frequencies, AC current flows only on conductor surface. Effective resistance increases with frequency. At 1 MHz, copper wire R is 100× higher than DC! Forces RF engineers to use thicker wires or special conductors.

Dry skin: 100 kΩ. Wet skin: 1 kΩ. Internal body: ~300 Ω. That's why electric shocks are deadly in bathrooms. 120 V across wet skin (1 kΩ) = 120 mA current—lethal. Same voltage, dry skin (100 kΩ) = 1.2 mA—tingle.

E12 series (10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82) covers each decade in ~20% steps. E24 series gives ~10% steps. E96 gives ~1%. Based on geometric progression, not linear—genius invention by electrical engineers!

Georg Ohm publishes V = IR. Ohm's law describes resistance quantitatively. Initially rejected by German physics establishment as 'web of naked fancies.'

British Association adopts 'ohm' as unit of resistance. Defined as resistance of mercury column 106 cm long, 1 mm² cross-section at 0°C.

First International Electrical Congress defines practical ohm. Legal ohm = 10⁹ CGS units. Named after Georg Ohm (25 years after his death).

International Electrical Congress adopts 'mho' (ohm backwards) for conductance. Later replaced by 'siemens' in 1971.

Heike Kamerlingh Onnes liquefies helium. Enables low-temperature physics experiments. Discovers superconductivity in 1911 (zero resistance).

Superconductivity discovered! Mercury resistance drops to zero below 4.2 K. Revolutionizes understanding of resistance and quantum physics.

Quantum Hall effect discovered. Resistance quantized in units of h/e² ≈ 25.8 kΩ. Provides ultra-precise resistance standard (accurate to 1 part in 10⁹).

SI redefinition: ohm now defined from fundamental constants (elementary charge e, Planck constant h). 1 Ω = (h/e²) × (α/2) where α is fine structure constant.

Resistance (R) opposes current flow, measured in ohms (Ω). Conductance (G) is the reciprocal: G = 1/R, measured in siemens (S). High resistance = low conductance. They describe the same property from opposite perspectives. Use resistance for series circuits, conductance for parallel (easier math).

In metals, electrons flow through a crystal lattice. Higher temperature = atoms vibrate more = more collisions with electrons = higher resistance. Typical metals have +0.3 to +0.6% per °C. Copper: +0.39%/°C. This is the 'positive temperature coefficient.' Semiconductors have opposite effect (negative coefficient).

Use reciprocals: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃... For two equal resistors: R_total = R/2. Easier method: use conductance! G_total = G₁ + G₂ (just add). Then R_total = 1/G_total. For example: 10 kΩ and 10 kΩ in parallel = 5 kΩ.

Tolerance = manufacturing variation (±1%, ±5%). Fixed error at room temp. Temperature coefficient (tempco) = how much R changes per °C (ppm/°C). 50 ppm/°C means 0.005% change per degree. Both matter for precision circuits. Low-tempco resistors (<25 ppm/°C) for stable operation.

E12 series uses ~20% steps in geometric progression. Each value is ≈1.21× previous (12th root of 10). This ensures uniform coverage across all decades. With 5% tolerance, adjacent values overlap. Brilliant design! E24 (10% steps), E96 (1% steps) use same principle. Makes voltage dividers and filters predictable.

In passive components, no—resistance is always positive. However, active circuits (op-amps, transistors) can create 'negative resistance' behavior where increasing voltage decreases current. Used in oscillators, amplifiers. Tunnel diodes naturally show negative resistance in certain voltage ranges. But true passive R > 0 always.