Resistance is the measure of how much a specific physical object opposes electrical current, while resistivity is the intrinsic material property that dictates how strongly that substance resists current regardless of its shape or size. In practical circuit design, resistance determines your actual voltage drop and heat generation, whereas resistivity tells you which material to choose before you even cut a wire or route a trace. People commonly confuse resistance with impedance (which includes AC reactance from capacitors and inductors) or conductance (the mathematical inverse of resistance, measured in Siemens).
The Core Difference: Material Property vs. Object Property
To understand the distinction, use this single analogy: imagine water flowing through a garden hose. The overall restriction to water flow depends on the hose's length and diameter (resistance). However, the inherent friction caused by the rubber or plastic lining of the hose itself is a fixed property of that material (resistivity). You can change the hose's resistance by cutting it shorter, but you cannot change the rubber's resistivity without swapping it for a different material entirely.
| Feature | Resistivity ($\rho$) | Resistance ($R$) |
|---|---|---|
| Definition | Intrinsic opposition to current flow of a material | Total opposition to current flow of a specific object |
| Symbol | $\rho$ (Greek letter rho) | $R$ |
| Unit | Ohm-meters ($\Omega\cdot\text{m}$) | Ohms ($\Omega$) |
| Depends on Geometry? | No (Material specific) | Yes (Length, cross-sectional area) |
| Formula | $\rho = R \cdot \frac{A}{L}$ | $R = \rho \cdot \frac{L}{A}$ |
According to data compiled by The Engineering Toolbox, annealed copper at 20°C has a resistivity of roughly $1.68 \times 10^{-8} \, \Omega\cdot\text{m}$. This number remains constant whether you have a microscopic fleck of copper or a massive busbar. However, the resistance of that copper changes drastically depending on how you shape it.
Worked Numeric Example: 12 AWG Copper vs. Aluminum Voltage Drop
Let's look at how resistivity dictates real-world wire sizing and voltage drop. Suppose you are wiring a 120V AC branch circuit to a workshop subpanel, requiring a one-way run of 50 meters (100 meters total for the hot and neutral loop). You are pulling a continuous 15A load and deciding between 12 AWG copper and 12 AWG aluminum wire.
Step 1: Identify the geometric constants.
Per NEC Chapter 9, Table 8, a solid 12 AWG wire has a cross-sectional area ($A$) of $3.31 \, \text{mm}^2$, or $3.31 \times 10^{-6} \, \text{m}^2$. Our total loop length ($L$) is 100 m.
Step 2: Calculate Resistance for Copper.
Copper resistivity ($\rho_{Cu}$) = $1.68 \times 10^{-8} \, \Omega\cdot\text{m}$.
$R_{Cu} = \rho_{Cu} \times \frac{L}{A} = (1.68 \times 10^{-8}) \times \frac{100}{3.31 \times 10^{-6}} = 0.507 \, \Omega$.
Step 3: Calculate Resistance for Aluminum.
Aluminum resistivity ($\rho_{Al}$) = $2.65 \times 10^{-8} \, \Omega\cdot\text{m}$.
$R_{Al} = \rho_{Al} \times \frac{L}{A} = (2.65 \times 10^{-8}) \times \frac{100}{3.31 \times 10^{-6}} = 0.800 \, \Omega$.
Step 4: Determine Voltage Drop and Power Loss.
Using Ohm's Law ($V = I \times R$) at 15A:
Aluminum Voltage Drop: $15\text{A} \times 0.800\Omega = \mathbf{12.0\text{V}}$ (10.0% drop)
While the NEC generally recommends a maximum voltage drop of 3% for branch circuits (meaning both 12 AWG options are technically undersized for a 50m run at 15A), this example perfectly illustrates the penalty of higher resistivity. The aluminum wire dissipates $144\text{W}$ of heat ($I^2R$) across the run, compared to just $114\text{W}$ for the copper. To match the copper's resistance using aluminum, you would need to increase the cross-sectional area by roughly 60%, which is why aluminum feeders are always sized larger than copper equivalents.
Where You Meet This in Practice
You will encounter the interplay between resistivity and resistance across several common electrical and electronics tasks:
- Home Wiring and Ampacity: The ampacity tables in NEC Article 310 are essentially thermal limits based on resistance. When current flows through a wire's resistance, it generates heat ($I^2R$). If the wire's insulation (like THHN or NM-B) cannot dissipate that heat, it degrades. Higher resistivity materials generate more heat at the same current, requiring larger wire gauges to maintain safe operating temperatures.
- PCB Trace Routing: In printed circuit board design, copper thickness is measured in ounces per square foot (1 oz copper is approximately $35 \, \mu\text{m}$ thick). Because the thickness is fixed, you control the trace's resistance by altering its width. High-current paths (like a 5A motor drive) require wide traces to lower resistance, while signal lines can remain narrow since their current is negligible. Tools like the Saturn PCB Toolkit use copper's known resistivity to calculate exact trace widths needed to prevent voltage drop and overheating.
- Heating Elements: If you want to generate heat, you want high resistance. Instead of using highly conductive copper, manufacturers use Nichrome (an alloy of nickel and chromium), which has a resistivity roughly 65 times higher than copper ($1.10 \times 10^{-6} \, \Omega\cdot\text{m}$). This allows a reasonably sized, short coil to achieve the high resistance necessary to convert electrical energy into thermal energy efficiently.
- Sense Resistors (Shunts): For current measurement, you need a precise, known resistance. Shunt resistors are often made from Manganin or Constantantin. These alloys are chosen not just for their specific resistivity, but because their Temperature Coefficient of Resistance (TCR) is nearly zero, meaning their resistance won't drift as they heat up from the current they are measuring.
Frequently Asked Questions
Does temperature change resistivity and resistance equally?
Yes, because resistance is directly proportional to resistivity ($R = \rho \frac{L}{A}$), any temperature-induced change in the material's resistivity will cause an identical percentage change in the object's overall resistance. For most pure metals, resistivity increases linearly with temperature (a positive temperature coefficient, or PTC). For example, copper's resistance increases by about 0.39% for every 1°C rise. This is why a cold incandescent bulb draws a massive inrush current (low cold resistance) that drops significantly once the tungsten filament heats up to operating temperature (high hot resistance). Conversely, materials like silicon or carbon have a negative temperature coefficient (NTC), where resistivity drops as they get hotter.
Why is aluminum wire larger than copper for the same resistance?
Aluminum has a resistivity roughly 58% higher than copper ($2.65 \times 10^{-8} \, \Omega\cdot\text{m}$ vs $1.68 \times 10^{-8} \, \Omega\cdot\text{m}$). Looking at the resistance formula $R = \rho \frac{L}{A}$, if $\rho$ increases by 58%, the cross-sectional area ($A$) must also increase by 58% to keep $R$ constant. In standard AWG sizing, this usually means stepping up two wire sizes. For instance, to safely carry 100A, you might use 3 AWG copper, but you would need 1 AWG aluminum to achieve the same electrical and thermal performance. Aluminum is used in utility transmission and large feeders not because it is electrically superior, but because its lower density makes it lighter and cheaper per ampere over long distances.
How do I calculate PCB trace resistance using copper resistivity?
To calculate PCB trace resistance, you apply the standard formula using the specific dimensions of your trace. First, determine the cross-sectional area. If you are using standard 1 oz copper, the thickness ($t$) is $35 \, \mu\text{m}$ ($35 \times 10^{-6} \, \text{m}$). If your trace width ($w$) is $0.5 \, \text{mm}$ ($0.5 \times 10^{-3} \, \text{m}$), the area $A = t \times w = 1.75 \times 10^{-8} \, \text{m}^2$. For a trace length ($L$) of $10 \, \text{cm}$ ($0.1 \, \text{m}$), the resistance is $R = (1.68 \times 10^{-8} \times 0.1) / (1.75 \times 10^{-8}) = 0.096 \, \Omega$. While this seems small, if that trace carries 2A, it will drop nearly 200mV and dissipate 380mW of heat, which can cause localized hot spots on a densely packed board. For high-current designs, always use a proper trace width calculator or increase copper weight to 2 oz.
