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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.

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The complex impedance of an object is given by:

$$ Z = R + i\left(\omega L - \frac{1}{\omega C}\right) $$

where $L$ is the inductance of the object, $C$ is the capacitance, and $\omega$ is the angular frequency of the voltage. Since the impedance is a function of frequency you cannot simply define the equivalent of resistivity. You cannot even factor out the frequency since the frequency dependence of the inductance and capacitance is different.

In addition the inductance and capacitance are not only properties of the material because they depend on the shape of the object. For example a long straight wire has a very small (though not zero) inductance, but wind that wire into a coil and the inductance increases dramatically. Again this means there is no property of the material we can use to define the impedance.

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  • $\begingroup$ So are you saying that resistance is independent of frequency, and it is this that enables one to define a geometry independent resistivity? $\endgroup$
    – Chris
    Oct 28 '21 at 8:25
  • $\begingroup$ @Chris Yes. The resistance is independent of frequency. $\endgroup$ Oct 28 '21 at 8:30
  • $\begingroup$ The source microwaves101.com/encyclopedias/rf-sheet-resistance suggests that 'sheet resistance' is a function of frequency. However, it is only the size of the sheet that relates the sheet resistance to the resistance, so this feels like a contradiction. $\endgroup$
    – Chris
    Oct 28 '21 at 9:01
  • $\begingroup$ @Chris That's because a sheet has an inductance as well as a resistance. It is the inductive part of the sheet impedance that is frequency dependent. Under normal conditions the inductance of a sheet is too low to make much difference, but the inductive impedance is proportional to ω and microwaves are very high frequency. That's why in the article you link (about microwaves) the inductance matters, $\endgroup$ Oct 28 '21 at 9:06
  • $\begingroup$ @Chris a component has an inductance when it has an associated magnetic field due to the current passing through it. The inductance is due to energy transfer to and from that magnetic field. Even a straight wire has a magnetic field when current passes through it, but the field is normally too weak to make much difference. Wind the wire into a coli and now you have an electromagnet with a strong field, so it has a much higher inductance. So the sheet in the example you link will have an inductance albeit a small one. $\endgroup$ Oct 28 '21 at 9:12
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There isn't a formula that just depends on the material because the reactance $X$ also depends on the frequency of the A.C. supply.

For a capacitor $$X_C = \frac{1}{2\pi f C}$$ and for an inductor $$X_L = 2\pi f L$$

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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.

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The inductance and capacitance of a wire can't be related to material properties of the wire because these circuit effects depend more on the fields around the wire than on what's going on in the wire itself.

Therefore they depend more on the material properties of the material around the wire than on the properties of the wire.

The relevant material properties are the dielectric permittivity (often designated $\varepsilon$) for capacitive effects and magnetic permeability ($\mu$) for inductive effects.

Note: We can find a self-inductance for the wire itself, but this is rarely the dominant inductive effect in a circuit.

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