The assumed resistance formula—formally known as Pouillet's Law—calculates the theoretical DC resistance of a uniform conductor based on its material properties and physical dimensions. Before you ever touch a multimeter to a wire run or a PCB trace, this formula tells you what the resistance should be. The core equation is:
R = ρ (L / A)
Whether you are sizing a 200A subpanel feeder or calculating voltage drop on a 12V solar array, assuming the correct baseline resistance prevents undersized wire, excessive heat, and failed components. Below is the complete breakdown, reference data, and worked math to apply this on the bench or jobsite.
The Core Formula and Symbol Definitions
To use the formula correctly, you must match the units of resistivity to your length and area measurements. The most common unit trap in electrical work is mixing metric area (mm²) with imperial resistivity constants. The table below defines every variable and its standard practical units.
| Symbol | Parameter | Standard Practical Unit | Definition & Notes |
|---|---|---|---|
| R | Assumed Resistance | Ohms (Ω) | The calculated DC opposition to current flow at a specific reference temperature (usually 20°C / 68°F). |
| ρ (rho) | Electrical Resistivity | Ω·mm²/m or Ω·cmil/ft | An intrinsic material property. Lower values mean better conductivity. Must match the L and A units. |
| L | Length | Meters (m) or Feet (ft) | The total one-way physical length of the conductor. For voltage drop, remember to double this for the out-and-back loop. |
| A | Cross-Sectional Area | mm² or Circular Mils (cmil) | The solid geometric area of the wire. For stranded wire, this is the sum of the individual strand areas, not the outer diameter. |
Material Resistivity Reference Data
You cannot calculate assumed resistance without the correct ρ value. The table below provides the resistivity of common conductors at the standard reference temperature of 20°C (68°F). We include both metric and imperial constants to prevent unit-conversion errors in the field. For deeper material science context, refer to the All About Circuits guide on factors affecting resistance.
| Material | Metric ρ (Ω·mm²/m) | Imperial ρ (Ω·cmil/ft) | Common Applications |
|---|---|---|---|
| Annealed Copper | 0.01724 | 10.37 | Standard branch circuits, PCB traces, magnet wire. |
| Aluminum (EC Grade) | 0.0282 | 17.0 | Service entrance feeders, utility transmission lines. |
| Tungsten | 0.056 | 33.8 | Incandescent lamp filaments, high-temp vacuum environments. |
| Constantan | 0.49 | 295 | Thermocouples, precision shunt resistors (low tempco). |
| Nichrome 80 | 1.08 | 650 | Toaster heating elements, high-wattage dummy loads. |
Rearranged Forms and Realistic Magnitudes
Depending on what you are designing or troubleshooting, you will need to isolate different variables. Here are the algebraic rearrangements of the assumed resistance formula:
- Solving for Resistivity: ρ = (R · A) / L
Use case: Identifying an unknown wire alloy by measuring its resistance and dimensions. - Solving for Length: L = (R · A) / ρ
Use case: Calculating the maximum run length before a specific voltage drop threshold is breached. - Solving for Area: A = (ρ · L) / R
Use case: Sizing a wire gauge to keep resistance below a target limit for a high-current DC load.
What a Realistic Answer Magnitude Looks Like
A major troubleshooting skill is knowing when your meter is lying to you. If your calculated assumed resistance is in the milliohm range, but your multimeter reads 2 Ω, you have probe contact resistance or a blown internal fuse in the meter. Here are realistic baseline magnitudes at 20°C:
- 22 AWG Hookup Wire: ~53 mΩ per meter.
- 12 AWG THHN Copper: ~1.6 mΩ per foot.
- 2/0 AWG Feeder Cable: ~0.16 mΩ per meter.
- 1 oz PCB Trace (10 mil width): ~50 mΩ per inch.
Worked Examples with Strict Unit Tracking
The most common point of failure in these calculations is dropping a unit conversion. Below are two step-by-step problems demonstrating strict unit tracking for both metric and imperial systems. For more on the physics underlying these calculations, see the Khan Academy module on resistance and resistivity.
Problem 1: Metric Solar Array Wire Sizing
Scenario: You are wiring a 12V DC solar charge controller. The one-way run from the panels to the controller is 15 meters. You are using standard copper wire (2.5 mm²). What is the assumed resistance of the one-way run?
- Identify Knowns: ρ = 0.01724 Ω·mm²/m; L = 15 m; A = 2.5 mm².
- Select Formula: R = ρ (L / A)
- Substitute Values: R = 0.01724 · (15 / 2.5)
- Calculate Area/Length Ratio: 15 m / 2.5 mm² = 6 m/mm²
- Final Multiply: 0.01724 Ω·mm²/m · 6 m/mm² = 0.10344 Ω (or ~103 mΩ).
Practical Note: Because current must return to the source, the total loop resistance is double this value (0.207 Ω). At 10A, this yields a voltage drop of roughly 2.07V, which is too high for a 12V system. You would need to rearrange for A and select a 6 mm² wire.
Problem 2: Imperial Branch Circuit Voltage Drop
Scenario: You are pulling a 100-foot run of 10 AWG solid copper wire to a 240V baseboard heater. What is the assumed resistance of this single conductor?
- Identify Knowns: ρ = 10.37 Ω·cmil/ft; L = 100 ft.
- Lookup Area: Per the NEC Chapter 9 Table 8, 10 AWG has a cross-sectional area of 10,380 circular mils (cmil).
- Select Formula: R = ρ (L / A)
- Substitute Values: R = 10.37 · (100 / 10,380)
- Calculate: R = 10.37 · 0.0096339 = 0.0999 Ω (essentially 0.1 Ω).
Boundary Conditions: Assumptions and Unit Traps
The assumed resistance formula is a DC, steady-state, ideal geometry model. If you apply it blindly to real-world AC mains or high-temperature environments, your calculations will fail. Keep these boundary conditions in mind:
The ρ values provided above are strictly for 20°C (68°F). Copper's resistance increases by approximately 0.39% for every 1°C rise in temperature. If your THHN wire is operating at its 75°C rating inside a hot attic, the actual resistance will be roughly 22% higher than the assumed formula predicts. Always multiply your assumed R by 1.22 for worst-case 75°C voltage drop calculations.
When the Formula Applies
- Uniform Cross-Sections: The wire or trace must have a constant area along its entire length. It fails for tapered grounds or crimped transitions.
- DC and Low-Frequency AC: At 50/60Hz, the formula is highly accurate for wire sizes up to roughly 1/0 AWG.
- Homogeneous Alloys: It assumes the material is pure. Copper-clad aluminum (CCA) wire cannot be calculated with this formula without complex parallel-resistor modeling of the core and the cladding.
Unit Mistakes That Break the Math
- The AWG / mm² Mix-up: Using a metric resistivity constant (0.01724) with an AWG wire size without converting the AWG to mm² first. Always convert AWG to mm² or use the circular mil constant.
- Stranding Factor Ignorance: Stranded wire has a slightly longer physical path than its linear jacket length due to the helical lay of the strands. This increases the assumed resistance by 1% to 2%. For precision shunt design, use solid core or apply a 1.02 multiplier to L.
- Confusing Diameter with Area: Plugging the wire diameter (e.g., 2.5mm) directly into A. You must calculate the area (A = π · r²) or look up the standardized cross-sectional area for the wire gauge.
By anchoring your wire sizing, PCB layout, and troubleshooting expectations to the assumed resistance formula, you eliminate guesswork. Measure the physical dimensions, apply the correct temperature-adjusted resistivity constant, and let the math dictate the wire gauge—not the other way around.
