Resistance Formula

Calculate resistance using resistivity, length, and area.

Inputs

\rho

Ω·m

L

A

m²

Formula Interpretation

Series Resistance

Parallel Resistance

Mixed Resistance

Resistance Formula

R = \rho \dfrac{L}{A}

$R$ (Ω) is resistance, $\rho$ (Ω·m) is material resistivity, $L$ (m) is conductor length, and $A$ (m²) is cross-sectional area. Resistance is proportional to length and inversely proportional to area — the fundamental relationship for wire and cable design. Reference resistivities: copper $1.72 \times 10^{-8}$ Ω·m, aluminium $2.82 \times 10^{-8}$ Ω·m, iron $1.0 \times 10^{-7}$ Ω·m.

Series Resistance

R = \sum_{i=1}^{n} R_i

For a pure series circuit, total resistance equals the sum of all branch resistances. Current is identical through each resistor, while voltage drops add.

Parallel Resistance

\dfrac{1}{R} = \sum_{i=1}^{n} \dfrac{1}{R_i}

For a pure parallel circuit, conductances add, so the reciprocal resistance form is used. The equivalent resistance is always less than the smallest branch resistance.

Mixed Series-Parallel

\begin{aligned} R &= R_{\parallel} + \sum_{j=1}^{m} R_{s,j} \\ \dfrac{1}{R_{\parallel}} &= \sum_{i=1}^{n}\dfrac{1}{R_{p,i}} \end{aligned}

For the shown mixed network, first reduce the parallel part of $R_1$ and $R_2$ , then add the series resistor $R_3$ to get the total equivalent resistance.

Knowledge Points

Resistivity and material classification

Resistivity $\rho$ is an intrinsic material property spanning roughly 25 orders of magnitude: conductors (copper $\approx 10^{-8}$ Ω·m) → semiconductors (silicon $\approx 10^{3}$ Ω·m) → insulators (PTFE $\approx 10^{16}$ Ω·m). For the same geometry, copper's resistance is about 61% that of aluminium — making it the preferred choice for high-conductivity applications. Aluminium offers lower density and cost, making it standard for overhead transmission lines and automotive wiring harnesses.

Effect of geometry on resistance

Resistance scales linearly with length $L$ and inversely with area $A$ . Doubling the cross-sectional area halves the resistance and increases current capacity by approximately 41% (since $I \propto \sqrt{A}$ at a given power loss). In practice, increasing conductor area directly reduces both voltage drop and $I^2R$ heat loss; shortening the run length (e.g. optimising PCB layout) is equally effective without adding weight.

Temperature effect and temperature coefficient

Metal resistivity rises with temperature; the corrected value is $R(T) = R_0[1 + \alpha(T - T_0)]$ , where $\alpha$ is the temperature coefficient (copper $\approx 3.93 \times 10^{-3}$ /°C, aluminium $\approx 4.03 \times 10^{-3}$ /°C) and the reference temperature $T_0$ is usually 20°C. In contrast, NTC (Negative Temperature Coefficient) thermistors decrease in resistance as temperature rises — used for temperature sensing and inrush-current limiting. PTC elements increase resistance sharply above a threshold — used as resettable fuses.

Skin effect at high frequencies

At DC and low frequencies, current is uniformly distributed over the entire cross-section and $R = \rho L / A$ applies exactly. At high AC frequencies, the skin effect concentrates current near the conductor surface within the skin depth $\delta = \sqrt{\rho / (\pi f \mu)}$ . At 1 MHz the skin depth in copper is only about 66 μm — far less than the conductor radius for typical wire — so the effective current-carrying area shrinks dramatically and $R_{\text{AC}} \gg R_{\text{DC}}$ . RF and switching-power-supply designers use multi-strand Litz wire or hollow conductors to recover the lost area and reduce AC resistance.

Example

For a copper wire with $\rho = 1.7 \times 10^{-8}\,\Omega\cdot\text{m}$ , length $L = 10\,\text{m}$ , and area $A = 1\,\text{mm}^2$ , compute the resistance.

Step 1 — Compute resistance

R = \rho \dfrac{L}{A} = 1.7 \times 10^{-8} \dfrac{10}{1 \times 10^{-6}} \approx 0.17\,\Omega

Step 2 — Parallel/series comparison

R_{\parallel} = \dfrac{R}{2} = 0.085\,\Omega

R_{\text{series}} = 2R = 0.34\,\Omega

Therefore the resistance is $R \approx 0.17\,\Omega$ , and two identical wires give $R_{\parallel} = 0.085\,\Omega$ in parallel or $R_{\text{series}} = 0.34\,\Omega$ in series.

Extended Knowledge

•Conductivity and conductance：Conductance $G = 1/R$ (siemens, S) is the reciprocal of resistance; conductivity $\sigma = 1/\rho$ (S/m) is the reciprocal of resistivity. The International Annealed Copper Standard (IACS) defines pure annealed copper conductivity as 100% IACS ( $5.8 \times 10^7$ S/m); other conductors are rated relative to this benchmark (aluminium ≈ 61% IACS). Using conductance (admittance) matrices instead of resistance matrices often simplifies nodal analysis of large electrical networks.
•Wire gauge standards and resistance per unit length：The American Wire Gauge (AWG) and IEC 60228 standards classify conductors by cross-sectional area. AWG numbers decrease as area increases (AWG 10 ≈ 5.26 mm², AWG 24 ≈ 0.205 mm²); IEC common sizes are 1.5, 2.5, 4, 6, 10 mm², etc. A 2.5 mm² copper conductor at 20°C has a resistance of about 7.4 mΩ/m. A 400 m return loop (800 m total) therefore presents ≈ 5.9 Ω of line resistance — causing a ≈ 59 V drop at 10 A that must be verified against the permissible voltage deviation.
•Voltage drop and power loss in power cables：Cable resistance causes a line voltage drop $\Delta U = I \cdot R_{\text{line}}$ and a copper loss $P_{\text{loss}} = I^2 R_{\text{line}}$ , both critical constraints in power distribution design. Most electrical codes limit the allowable voltage deviation at the load end (typically ±5% of nominal). Increasing conductor area or reducing run length directly lowers both; high-voltage transmission (10 kV, 35 kV, 110 kV) reduces current and hence resistive loss in proportion to the square of the voltage ratio.
•PCB copper trace resistance estimation：PCB trace resistance is calculated as $R = \rho L / (W \cdot t)$ , where $W$ is trace width (m) and $t$ is copper thickness (standard 1 oz copper is $t \approx 35$ μm). For example, a 0.5 mm wide, 100 mm long 1 oz trace has $R \approx 97$ mΩ, producing a 97 mV drop and 97 mW of heat at 1 A. The IPC-2221 standard provides current-vs-width tables for both internal and external layers at various temperature-rise limits and is the authoritative reference for PCB current-carrying capacity design.

Resistance Formula

Calculate resistance using resistivity, length, and area.

Inputs

\rho

Ω·m

L

A

m²

Formula Interpretation

Series Resistance

Parallel Resistance

Mixed Resistance

Resistance Formula

R = \rho \dfrac{L}{A}

Series Resistance

R = \sum_{i=1}^{n} R_i

For a pure series circuit, total resistance equals the sum of all branch resistances. Current is identical through each resistor, while voltage drops add.

Parallel Resistance

\dfrac{1}{R} = \sum_{i=1}^{n} \dfrac{1}{R_i}

For a pure parallel circuit, conductances add, so the reciprocal resistance form is used. The equivalent resistance is always less than the smallest branch resistance.

Mixed Series-Parallel

\begin{aligned} R &= R_{\parallel} + \sum_{j=1}^{m} R_{s,j} \\ \dfrac{1}{R_{\parallel}} &= \sum_{i=1}^{n}\dfrac{1}{R_{p,i}} \end{aligned}

For the shown mixed network, first reduce the parallel part of $R_1$ and $R_2$ , then add the series resistor $R_3$ to get the total equivalent resistance.

Knowledge Points

Resistivity and material classification

Effect of geometry on resistance

Temperature effect and temperature coefficient

Skin effect at high frequencies

Example

For a copper wire with $\rho = 1.7 \times 10^{-8}\,\Omega\cdot\text{m}$ , length $L = 10\,\text{m}$ , and area $A = 1\,\text{mm}^2$ , compute the resistance.

Step 1 — Compute resistance

R = \rho \dfrac{L}{A} = 1.7 \times 10^{-8} \dfrac{10}{1 \times 10^{-6}} \approx 0.17\,\Omega

Step 2 — Parallel/series comparison

R_{\parallel} = \dfrac{R}{2} = 0.085\,\Omega

R_{\text{series}} = 2R = 0.34\,\Omega

Therefore the resistance is $R \approx 0.17\,\Omega$ , and two identical wires give $R_{\parallel} = 0.085\,\Omega$ in parallel or $R_{\text{series}} = 0.34\,\Omega$ in series.

Extended Knowledge

•Conductivity and conductance：Conductance $G = 1/R$ (siemens, S) is the reciprocal of resistance; conductivity $\sigma = 1/\rho$ (S/m) is the reciprocal of resistivity. The International Annealed Copper Standard (IACS) defines pure annealed copper conductivity as 100% IACS ( $5.8 \times 10^7$ S/m); other conductors are rated relative to this benchmark (aluminium ≈ 61% IACS). Using conductance (admittance) matrices instead of resistance matrices often simplifies nodal analysis of large electrical networks.
•Wire gauge standards and resistance per unit length：The American Wire Gauge (AWG) and IEC 60228 standards classify conductors by cross-sectional area. AWG numbers decrease as area increases (AWG 10 ≈ 5.26 mm², AWG 24 ≈ 0.205 mm²); IEC common sizes are 1.5, 2.5, 4, 6, 10 mm², etc. A 2.5 mm² copper conductor at 20°C has a resistance of about 7.4 mΩ/m. A 400 m return loop (800 m total) therefore presents ≈ 5.9 Ω of line resistance — causing a ≈ 59 V drop at 10 A that must be verified against the permissible voltage deviation.
•Voltage drop and power loss in power cables：Cable resistance causes a line voltage drop $\Delta U = I \cdot R_{\text{line}}$ and a copper loss $P_{\text{loss}} = I^2 R_{\text{line}}$ , both critical constraints in power distribution design. Most electrical codes limit the allowable voltage deviation at the load end (typically ±5% of nominal). Increasing conductor area or reducing run length directly lowers both; high-voltage transmission (10 kV, 35 kV, 110 kV) reduces current and hence resistive loss in proportion to the square of the voltage ratio.
•PCB copper trace resistance estimation：PCB trace resistance is calculated as $R = \rho L / (W \cdot t)$ , where $W$ is trace width (m) and $t$ is copper thickness (standard 1 oz copper is $t \approx 35$ μm). For example, a 0.5 mm wide, 100 mm long 1 oz trace has $R \approx 97$ mΩ, producing a 97 mV drop and 97 mW of heat at 1 A. The IPC-2221 standard provides current-vs-width tables for both internal and external layers at various temperature-rise limits and is the authoritative reference for PCB current-carrying capacity design.