# Scale-covariant decomposition of capacitance

I'm wondering if there is any good insight of how to evaluate a given capacitive geometry in such a way that it would be expressed as a function that depends only on two components:

• as a geometric component $C_g$, that depends on the shape of the capacitor

• as a scale-dependant component, $C_s$. In particular, that would depend on square laws on capacitive surfaces. Or in the case of a surface approximating some fractal surfaces, with some weird non-integral exponent between 2 and 3, like seems to be the case for Graphene capacitors

In other words, the capacitance expressed as a function of two parameters:

$$C = C( \textbf{g} , L )$$

where $\textbf{g}$ represents the scale-free geometric information of the capacitor, and L the scale parameter

Is there an easy physical intuition that allows to roughly predict what shape might be optimal for a capacitor at a given scale? is the one with the most folds up to the minimal allowed scale? does it matter how nested the folds are, or the structure of the nesting?

And more interestingly, what properties of the capacitor geometry makes it worse or better capacitor than the same geometry at a different scale?

• The last few comments got a little rough, so I deleted them to back up the conversation to a useful point. The rest of them have been moved to chat. – David Z Sep 3 '15 at 15:15