Resistance Principles
Part of Basic Electrical Circuits
The physical basis of electrical resistance—why materials resist current, how geometry and temperature affect resistance, and how to calculate resistance for any conductor.
Why This Matters
Electrical resistance is not a mysterious property—it has a clear physical explanation rooted in the atomic structure of materials. Understanding why resistance exists, and what factors increase or decrease it, allows a person to make informed decisions about conductor selection, to design circuits for specific resistance values, and to predict how circuits will behave when temperature changes.
More practically: resistance is responsible for every instance of useful heat generation (electric heaters, electric ovens, electric lamps) and every instance of unwanted energy waste (wiring that gets warm under load, batteries that discharge faster than expected). Understanding resistance means understanding which heating is useful and which is waste, and how to maximize one while minimizing the other.
The same physics explains why thick cables are used for high-current power distribution, why tungsten filaments glow instead of burning, and why long wire runs in a building have measurable voltage drops. All of these follow directly from resistance principles.
The Atomic Model of Resistance
Metals conduct electricity because their outer electrons are loosely bound and can move freely throughout the material—the “electron sea” or “free electron” model. When a voltage is applied, these free electrons drift toward the positive terminal, constituting an electric current.
This drift is not unimpeded. The electrons collide with:
- Ion cores of the metal atoms in the crystal lattice
- Lattice defects (vacant sites, displaced atoms, grain boundaries)
- Impurity atoms of other elements mixed into the metal
- Phonons (vibrations of the crystal lattice, which increase with temperature)
Each collision transfers energy from the electron to the lattice—the electron loses momentum, scatters, and must be reaccelerated by the electric field. This energy transfer is what heats the conductor. The more collisions per unit length, the higher the resistance.
Why temperature increases resistance in metals: Higher temperature means more vigorous lattice vibrations, which means more frequent collisions, which means higher resistance. Most metals increase in resistance by about 0.3–0.5% per degree Celsius rise. This effect is significant—a tungsten filament at 3000°C has resistance roughly 10× higher than at room temperature.
Why some alloys have near-zero temperature coefficient: Alloys like manganin and constantan are designed so that competing effects nearly cancel—the resistance change from temperature is less than 0.01% per °C. This makes them ideal for precision resistors and measurement shunts.
The Resistivity Formula
The resistance of any uniform conductor depends on three factors:
R = ρ × L / A
Where:
- R = resistance (ohms, Ω)
- ρ = resistivity (a material property, in Ω·m)
- L = length (meters)
- A = cross-sectional area (square meters)
Physical interpretation:
- Doubling the length doubles resistance (proportional): a longer path means more collisions
- Doubling the cross-sectional area halves resistance (inversely proportional): more parallel paths share the current, reducing collisions per electron
Resistivity values (at 20°C):
| Material | Resistivity (Ω·m) |
|---|---|
| Silver | 1.59 × 10⁻⁸ |
| Copper | 1.68 × 10⁻⁸ |
| Aluminum | 2.65 × 10⁻⁸ |
| Iron | 9.71 × 10⁻⁸ |
| Nichrome | 1.10 × 10⁻⁶ |
| Carbon (graphite) | 3–60 × 10⁻⁵ |
| Sea water | 0.2 |
| Glass | 10⁹–10¹³ |
| Rubber | 10¹³ |
The range from copper to rubber spans 20 orders of magnitude—one of the largest ranges of any material property in nature.
Calculating Resistance of Common Conductors
Copper wire: Cross-sectional area of a circular conductor: A = π(d/2)² = π × r²
For 1.5mm diameter copper wire: A = π × (0.00075)² = 1.77 × 10⁻⁶ m² R per meter = ρ/A = (1.68 × 10⁻⁸) / (1.77 × 10⁻⁶) = 0.0095 Ω/m = 9.5 mΩ/m
Resistance of 50m of 1.5mm copper wire: R = 0.0095 × 50 = 0.475Ω (one conductor) Round trip (both conductors): 0.95Ω
Nichrome heating element: To make a 20Ω heating element from 0.5mm diameter nichrome wire: A = π × (0.00025)² = 1.96 × 10⁻⁷ m² R/m = (1.10 × 10⁻⁶) / (1.96 × 10⁻⁷) = 5.6 Ω/m Length needed = 20Ω / 5.6Ω/m = 3.57 meters
Temperature Coefficient of Resistance
The resistance of a conductor at temperature T is: R_T = R_20 × [1 + α × (T - 20)]
Where:
- R_20 = resistance at 20°C (room temperature reference)
- α = temperature coefficient of resistance (per °C)
- T = operating temperature (°C)
Temperature coefficients:
| Material | α (per °C) |
|---|---|
| Copper | +0.0039 |
| Aluminum | +0.0039 |
| Iron | +0.005 |
| Nichrome | +0.0004 |
| Carbon | −0.0005 |
| Manganin | ±0.00002 |
Note: Carbon has a negative temperature coefficient—it becomes a better conductor as it heats up. This is characteristic of semiconductors and non-metallic conductors.
Practical example: A copper motor winding reads 5Ω at 20°C. After running for an hour, the resistance measures 6.3Ω. What is the winding temperature?
T = 20 + (R_T/R_20 - 1)/α = 20 + (6.3/5 - 1)/0.0039 = 20 + (0.26/0.0039) = 20 + 66.7 = 86.7°C
This technique—measuring winding temperature from resistance—is used for motor thermal protection. If the temperature exceeds safe limits, the motor should be shut down to prevent insulation damage.
Resistance of Conductors in Series and Parallel
Series: Total resistance = sum of all individual resistances R_total = R₁ + R₂ + R₃ + …
Parallel: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + …
These are consequences of geometry, not material properties. Series connection extends the path length; parallel connection increases the effective cross-sectional area.
Contact Resistance
The resistance of a metal surface is theoretically near zero—but in practice, the contact between two conductors is never perfectly clean and smooth. Oxide films, surface roughness, and contamination create contact resistance that can range from milliohms (clean silver) to kilohms (corroded iron).
Contact resistance consequences:
- A 1Ω contact resistance in a 10A circuit dissipates 100W of heat at that point—enough to cause fires
- A 0.1Ω contact resistance causes 1V drop in a 10A circuit—significant in 12V systems
- Varying contact resistance causes intermittent faults that are difficult to diagnose
Maintaining low contact resistance:
- Clean contact surfaces with fine abrasive before assembly
- Use conductive compound (petroleum jelly with graphite powder, or commercial anti-oxidant compound) on aluminum connections
- Silver-plate copper contacts in high-reliability applications
- Tighten all mechanical connections firmly; recheck periodically as thermal cycling loosens fasteners
Contact resistance is the most common cause of unexplained electrical failures. Any suspicious circuit should have all connections checked before replacing components.