Resistance and Resistivity

Resistance is defined as $R=\frac{V}{I}$, measured in ohms. This definition always holds — it is how resistance is defined, and it applies to any component at any instant.

$\text{Ohm’s law}$ is a separate and stronger claim: that for some components the current is $\text{proportional}$ to the potential difference, so $R$ stays constant. That holds only for $\text{ohmic}$ conductors at constant temperature.

The distinction matters. You can always compute $V/I$ for a lamp at a given point, but the answer changes with the operating point, so the lamp does not obey Ohm’s law.

Resistance depends on the object’s dimensions as well as its material. A longer wire has more resistance because carriers travel further against opposition; a thicker wire has less, because there are more parallel paths.

Those dependences combine into $R=\frac{\rho L}{A}$, where $\rho$ is the $\text{resistivity}$ in ohm-metres — an intrinsic property of the material, independent of shape.

Resistivity spans an extraordinary range: about $1.7\times 10^{-8}\Omega \mathrm{m}$ for copper and around $10^{16}\Omega \mathrm{m}$ for a good insulator — twenty-four orders of magnitude, which is why the same p.d. produces wildly different currents.

A $\text{metallic conductor at constant temperature}$ gives a straight line through the origin. Constant gradient means constant resistance, so it is ohmic.

A $\text{filament lamp}$ gives a curve that flattens as current increases. Higher current heats the filament, the metal ions vibrate more, electrons collide with them more often, and the resistance rises. The gradient falls because resistance is increasing.

A $\text{semiconductor diode}$ conducts in one direction only. Below the threshold voltage — about 0.6 V for silicon — almost no current flows; above it the current rises steeply. In reverse bias the resistance is extremely high and the current is negligible.

For a $\text{metal}$, resistance increases with temperature. The number of carriers is essentially fixed, but hotter ions vibrate more and scatter electrons more effectively, reducing the drift velocity for a given field.

For a $\text{negative temperature coefficient thermistor}$ — a semiconductor — resistance $\text{decreases}$ sharply with temperature. Heating releases many more charge carriers, and that increase in $n$ overwhelms the extra scattering. Referring back to $I=nAvq$ makes the contrast clear: metals have fixed $n$ and falling $v$; semiconductors have rapidly rising $n$.

Below a $\text{critical temperature}$, some materials become $\text{superconductors}$ with exactly zero resistivity. Current then flows with no energy dissipation, which is what makes powerful electromagnets in MRI scanners and particle accelerators practical.