The resistivity of a wire depends strictly on its base material and its operating temperature, defining the intrinsic opposition to electrical current flow independent of the wire's physical dimensions. When you are sizing feeders, calculating voltage drop, or troubleshooting a warm breaker panel, understanding this intrinsic property is the difference between a safe installation and a melted terminal lug. Before we pull any wire, we need to clear up the most common confusion on the workbench: resistivity ($\rho$, measured in ohm-meters) is a fundamental material property, while resistance ($R$, measured in ohms) is what you actually measure with your multimeter across a specific cut of wire. Resistivity is the recipe; resistance is the baked cake.
The Core Variables: Material and Temperature
When we ask what the resistivity of wire depends on, physics gives us two primary variables. The physical dimensions (length and cross-sectional area) dictate the final resistance, but the resistivity itself is locked to the atomic structure of the metal and how much thermal energy is agitating its lattice.
1. The Base Material
Every conductive metal has a specific atomic lattice that either facilitates or hinders the flow of free electrons. Silver has the lowest resistivity of any common metal, but its cost relegates it to specialized RF contacts and high-end audio. In residential and commercial wiring, we rely on copper and aluminum. According to Georgia State University's HyperPhysics database, the baseline resistivity of annealed copper at 20°C is $1.68 \times 10^{-8} \Omega\cdot m$, while aluminum sits at $2.82 \times 10^{-8} \Omega\cdot m$. This means aluminum inherently opposes current flow about 68% more than copper at room temperature.
2. Operating Temperature
Metals have a positive temperature coefficient of resistance. As the wire heats up—either from ambient attic temperatures or from $I^2R$ heating caused by the load itself—the metal lattice vibrates more violently, scattering electrons and increasing resistivity. For copper, the temperature coefficient ($\alpha$) is approximately 0.00393 per °C. This is why the NEC requires us to apply temperature correction factors (Table 310.15(B)(1)) when routing NM-B or THHN through hot environments.
Worked Example: Copper vs. Aluminum at 20°C and 75°C
Let's look at real numbers to see how temperature shifts the actual resistance of a circuit. We will calculate the resistance of a 100-foot (30.48-meter) one-way run of 12 AWG wire. The cross-sectional area of 12 AWG is $3.31 mm^2$ ($3.31 \times 10^{-6} m^2$). The formula for resistance is $R = \rho \times (L / A)$.
Copper (Cu) Calculation
- At 20°C (Room Temp): $R = (1.68 \times 10^{-8} \times 30.48) / (3.31 \times 10^{-6}) = 0.155 \Omega$
- At 75°C (Under Load): We apply the temperature shift formula $R_{hot} = R_{cold} \times [1 + \alpha(\Delta T)]$. With a $\Delta T$ of 55°C, the multiplier is $1 + (0.00393 \times 55) = 1.216$. The new resistance is $0.155 \times 1.216 = 0.188 \Omega$.
Aluminum (Al) Calculation
- At 20°C (Room Temp): $R = (2.82 \times 10^{-8} \times 30.48) / (3.31 \times 10^{-6}) = 0.259 \Omega$
- At 75°C (Under Load): Aluminum's temperature coefficient is slightly higher (~0.00429). The multiplier for a 55°C rise is 1.236. The new resistance is $0.259 \times 1.236 = 0.320 \Omega$.
This math proves exactly why the Copper Development Association and the NEC mandate sizing up aluminum conductors. A 12 AWG aluminum wire at operating temperature has more than 70% higher resistance than a 12 AWG copper wire, generating significantly more heat for the exact same 16A continuous load.
Where You Meet This in Practice
Theory is great for the classroom, but on the jobsite, what the resistivity of wire depends on translates directly into three practical scenarios:
1. Voltage Drop on Long Branch Circuits
Voltage drop is calculated as $V_{drop} = I \times R_{total}$. If you are running a 120V circuit to a detached garage 100 feet away using 12 AWG copper, the total wire length (out and back) is 200 feet. At 75°C, the total resistance is $0.376 \Omega$. If you pull a continuous 16A load (like a portable heater or a window AC unit), the voltage drop is $16A \times 0.376 \Omega = 6.01V$. That's a 5% drop on a 120V nominal circuit. The NEC recommends keeping branch circuit voltage drop under 3%. Because resistivity increases with temperature, a wire running warm will drop more voltage than the same wire running cold, potentially causing motors to overheat and draw even more current.
2. Attic Wiring and Ambient Derating
If you route NM-B (Romex) through an attic in the US Southwest, the ambient temperature can easily exceed 110°F (43°C) in the summer. The base resistivity of the copper climbs before the wire even carries a single amp of current. This is why NEC 310.15(B)(1) requires you to derate the ampacity of the wire. A 14 AWG wire normally rated for 15A might only be legally allowed to carry 12A in that environment because the elevated ambient resistivity pushes the insulation closer to its thermal failure point.
3. Aluminum Service Entrance Upgrades
When upgrading a home to a 200A service, electricians almost universally use 4/0 AWG aluminum instead of 2/0 AWG copper. While copper has lower resistivity, the cost per foot for 2/0 copper is prohibitive. By accepting aluminum's higher resistivity and compensating by increasing the cross-sectional area (using a thicker wire), we achieve the same total resistance and ampacity at a fraction of the material cost. The trade-off is that aluminum requires strict termination practices, including wire brushing and antioxidant compound, to prevent oxidation—which introduces a third, highly resistive material (aluminum oxide) into the circuit path.
Frequently Asked Questions
Does the resistivity of wire depend on its length or thickness?
No. This is the most common trap in electrical theory. Resistivity ($\rho$) is an intrinsic material property—it depends only on what the wire is made of and its temperature. Resistance ($R$), however, depends heavily on length and thickness. A 10-foot spool of 12 AWG copper and a 500-foot spool of 12 AWG copper have the exact same resistivity, but the 500-foot spool has 50 times the resistance.
Why does the resistivity of wire depend on temperature so heavily in attics?
As ambient temperature rises, the metal atoms in the wire vibrate with greater kinetic energy. This thermal agitation creates a 'rougher' path for free electrons to travel through, increasing the collision rate and thus the resistivity. In a 120°F attic, the copper is already starting at a higher baseline resistivity before your load even turns on, which is why the NEC mandates ampacity derating to prevent the insulation from melting.
How does alloying change what the resistivity of wire depends on?
Introducing impurities or alloying elements drastically increases resistivity. Pure, annealed copper is highly conductive, but it is soft. If you use brass or bronze (copper alloyed with zinc or tin) for a conductor, the foreign atoms disrupt the uniform copper lattice, scattering electrons and spiking the resistivity. This is why we use pure copper for wiring, but use high-resistivity alloys like Nichrome for heating elements in toasters and hair dryers.
What happens to circuit breakers if wire resistivity increases due to heat?
Circuit breakers monitor current (amperage), not resistance or resistivity. However, if a wire's resistivity increases due to extreme heat, the voltage drop across that wire increases. If the voltage at the appliance drops too low, devices with induction motors (like AC compressors or refrigerator pumps) will draw more current to maintain their mechanical power output ($P = V \times I$). This increased current draw can eventually trip the breaker, or worse, overheat the wire beyond its insulation rating if the breaker is oversized or fails to trip in time.
