What is the relationship between conductance and conductivity? Is there any formula that can explain their relationship?
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Conductance is the extrinsic property while conductivity is the intrinsic property. This means that conductance is the property of an object dependent of its amount/mass or physical shape and size, while conductivity is the inherent property of the material that makes up the object. No matter how the object changes in terms of shape/size/mass, as long as it is made of the same material and the temperature remains the same, its conductivity does not change. Conversely, the conductance of a conductor changes with its cross-sectional area and length. Of course, a higher conductivity also gives an object a higher conductance. The formula that relates conductivity with conductance is:
$$G=\sigma \frac A l$$
where $G$ is the conductance, $σ$ the conductivity, $A$ the cross-sectional area perpendicular to the direction of electric current, and $l$ the length of the conductor. This formula applies for any (geometrically) prismic or cylindrical conductor, including cuboids.
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$\begingroup$ I guess the formula is for the 3D case. How about for a 2D, 1D, or even 0D system? $\endgroup$ Commented Feb 19, 2021 at 1:49
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1$\begingroup$ @ΑλέξανδροςΖεγγ If your object is 2D or 1D (i.e. has zero thickness) there will be no actual material to conduct anything, so it would be 0. If you just have a very thin object with constant thickness then you can use formulas like $G = k\frac{\text{width}}{l}$ (where $k = \sigma \times \text{thickness}$) or $G = k \frac1{l}$ (where $k = \sigma A$), but really they are just the same formula with different arrangements of variables. At the end of the day you still have to either know the value of $A$ or experimentally measure the value of $k$. $\endgroup$ Commented Feb 19, 2021 at 2:57
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$\begingroup$ Then do the $ k $s in the two cases have a proper name? $\endgroup$ Commented Feb 19, 2021 at 3:02
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$\begingroup$ @ΑλέξανδροςΖεγγ and if your object is 0D, it's essentially a point object. There isn't even a length to conduct across, and the formula boils down to something like $G = k$ where $k = \sigma \frac{A}{l}$, in which case we shouldn't bother about conductivity, since the whole formula literally say "conductance G equals constant k", which basically means that we're directly assigning a value to the conductance rather than calculating anything, and there may as well be no formula. $\endgroup$ Commented Feb 19, 2021 at 3:06
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1$\begingroup$ @ΑλέξανδροςΖεγγ I've heard people call the resistivity versions "areal resistivity" and "linear resistivity", I suppose you can just replace "resistivity" with "conductivity", but it's not a very common term. If you preferred you could probably call it coefficient of conductance or something. $\endgroup$ Commented Feb 19, 2021 at 3:11