Resistance and Resistivity

Resistance — the opposition of a component to the flow of charge:

R = V / I

Measured in ohms. One ohm is one volt per ampere. This equation defines resistance and is always true; it is not a statement of Ohm's law.

Ohm's law — the current through a conductor is directly proportional to the potential difference across it, provided physical conditions such as temperature stay constant. A component that obeys this is ohmic, and its I–V graph is a straight line through the origin. Many components are not ohmic, which does not stop R = V/I from applying to them at any particular point.

Resistivity — a property of the material rather than of a particular piece of it:

R = ρL / A

Measured in ohm metres (Ω m). Resistance rises with length and falls with cross-sectional area, which is why long thin wires resist more than short thick ones.

Example: a 2.0 m wire of area 5.0 × 10⁻⁷ m² and resistivity 1.7 × 10⁻⁸ Ω m has R = (1.7 × 10⁻⁸ × 2.0) ÷ (5.0 × 10⁻⁷) = 0.068 Ω. Copper's resistivity is tiny, which is exactly why it is used for wiring.

Why resistivity rather than resistance — resistance depends on the dimensions of the sample, so quoting it tells you about that object. Resistivity strips out length and area and describes the substance, which is the same relationship that stress and strain have to force and extension.

Temperature and metals — heating a metal makes the positive ions vibrate with greater amplitude, so the free electrons collide with them more often and lose more energy. The number of charge carriers is essentially unchanged, so the resistance increases.

Measuring resistivity — measure the resistance of a known length of wire, and its diameter with a micrometer at several points. Better, vary the length and plot R against L: the gradient is ρ/A, and using a gradient averages random error and exposes a non-zero intercept caused by contact resistance.