The resistivity of a wire, $\rho$, is purely a material property. For a wire of length, $l$, and cross-sectional area, $A$, the resistance, $R$, is given by
$$
R=\frac{\rho l}{A}
$$
which depends on the geometry. Given that the impedance, $Z$, relates to the resistance and the reactance, $X$, via
$$
Z=R+iX
$$
is there a way of expressing/defining a 'reactivity' or 'impedivity' (made up I'm sure) of a wire, or a material more generally, that's only a material property? I have never heard of such expressions and I'm not sure why. Is there a reason why such expressions don't exist? When people talk about the impedance of a sheet, they assume it's infinitely thin, so where does the geometry come in? Is it purely a material property?
Thanks in advance for any help.
Best Answer
Ohm's law for materials is usually written as $$\mathbf{j}=\hat{\sigma}\mathbf{E},$$ where $\mathbf{j}$ is the current density and \mathbf{E} is the local electric field. $\hat{\sigma}$ in this case could be a tensor and coul be complex (in which case the relation applies for a particular frequency $$\mathbf{j}(\mathbf{x},\omega)=\hat{\sigma}(\omega)\mathbf{E}(\mathbf{x},\omega),$$ whereas in the time domain one would have to write $$\mathbf{j}(\mathbf{x},t)=\int_{-\infty}^{+\infty}d\tau\hat{\sigma}(t-\tau)\mathbf{E}(\mathbf{x},\tau).$$
$\hat{\sigma}(\omega)$ reflects local properties of the material, rather than those of the bulk conductor, which can be trivially obtained by integrating the current ove rthe full cross-section and relating the electric field to the potential difference (see, e.g., this derivation). Virtually any text dealing with Kubo formula focuses on calculating $\hat{\sigma}(\omega)$.
Remark: Describing resistivity via a local quantity works only withing the context of macroscopic electrodynamics/transport theory - as long as the quantum coherence length is much shorter than the dimension of the physically infenitesimal volume. When dealing with quantum nanostructures one cannot anymore separate the material properties and the geometry of the structure, as manifested bys uch phenomena as conductance quantization, quantum Hall effect, weak localization, etc. Joe Imry's Introduction to mesoscopic physics provides a good introduction into this field.