Ohm's second law: resistivity and geometry

Ohm’s first law introduces resistance $R$ as a property of the component, but does not say what it depends on. Ohm’s second law fills this gap: it shows how $R$ arises from the geometry of the conductor and the material it is made of.

Key formula — Ohm's second law

The resistance of a wire-shaped conductor depends on its geometry and material according to $\ev{R = \rho\,\frac{L}{S}}$ where $L$ is the length, $S$ the cross-sectional area and $\rho$ the material’s resistivity.

The physical meaning is intuitive. Resistance grows with the length $L$: a longer wire opposes more obstacles to the passage of charges, just as a longer pipe slows the flow down more. Resistance instead decreases as the cross-section $S$ increases: a thicker conductor offers more parallel “lanes” to the charges, just as a wider pipe lets more water flow through for the same pressure difference.

Resistivity

Resistivity $\rho$ is the quantity that intrinsically characterises the material: it depends only on the substance the conductor is made of (and, as we shall see, on temperature), not on its shape. Two wires of different materials but identical geometry will have different resistances precisely because they differ in $\rho$. From the formula we can derive its unit of measurement:

$\ev{[\,\rho\,] = \Omega\cdot\mathrm{m}}$

A low value of $\rho$ indicates a good conductor. The metals used for cables have very small resistivities:

Material	Resistivity $\rho$ ($\Omega\cdot$m)
Copper	$1{,}7\cdot10^{-8}$
Aluminium	$2{,}8\cdot10^{-8}$
Iron	$10^{-7}$

Copper is the conductor of choice in wiring installations owing to its low resistivity combined with acceptable cost and workability; aluminium, somewhat more resistive but much lighter, is preferred for overhead transmission lines; iron, with a resistivity almost ten times that of copper, is a distinctly worse conductor.

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Topics: Circuiti elettrici Concepts: Legge di Ohm

Related exercises: Problem — resistance of a light bulb · Problem — resistance and power of an ohmic conductor · Problem — voltage drop in a copper cable