= Solution
In a normal <metal>, the <electrical resistivity> can often be approximated using <Matthiessen's rule>,
$$
\rho_{\rm el}(T)\simeq\rho_{\rm residual}+\rho_{\rm phonon}(T).
$$
Static impurities, defects and grain boundaries scatter conduction <Electrons> even as $T\to0$, giving the <residual resistivity>. Thermal lattice <phonons> add a temperature-dependent contribution through <electron-phonon scattering>. In a clean three-dimensional acoustic model, <small-angle electron-phonon transport at low temperature> gives $\rho_{\rm phonon}\propto T^5$ at low $T$; once thermal <phonon> populations are approximately classical, the resistivity is approximately linear in $T$. An electron-electron contribution proportional to $T^2$ can be relevant in some low-temperature ranges.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2018/paper-2-c11-metal-resistivity.png]
{title=Schematic normal-metal resistivity with residual, low-temperature phonon and approximately linear high-temperature regimes}
<Carbon> in mild steel, together with associated disorder, strain and microstructural changes, adds scattering and generally raises the <residual resistivity> relative to purer <iron>. Thus a dilute alloy may have a similar broad phonon-dependent rise but a larger baseline. The sketch is a generic phonon-plus-disorder model, not measured <iron> data. <Iron>'s magnetic scattering and <phase> changes, and real steel's phases and processing history, can add structure; the simple sum is approximate rather than an exact material law. \b[The expected explanation is greater <Electron> scattering from the added <carbon> and resulting disorder.]
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