Derive formula for capacity of a wrapped capacitor from geometry Consider two conducting sheets separated by a dielectric and wound like in the picture below. Suppose the area of one of the unrolled sheets is $A$, the distance between the sheets is $d$ and the the dielectric $\epsilon_r$ fills the total space between the plates.
What is the capacitance of this system? If it was unrolled it would be $C = \epsilon_0 \epsilon_r A/d$. I think in wound configuration the capacitance should be larger by a factor 2 since then every plate "sees" each other twice (expect of boundary effects).
I feel that this argument is very vague and I am looking for a way to see this clearer in more detail. 

 A: You are correct. Rolling-up increases the capacitance of a parallel plate capacitor by a factor of up to 2 in the limit of many complete turns with a radius much larger than the separation of the plates.
When the parallel plate capacitor is not rolled up, only the facing plates can store charge. The outer plates are not used. When the capacitor is rolled up with an extra layer of dielectric the outer faces of the two plates are now adjacent to each other, forming a second capacitor in parallel with the inner plates. 
This is not true for all parts of the "roll." As you hint with regard to "boundary effects", the outermost and innermost layers have an adjacent plate on only one side, not both, so these cannot contribute to the factor of 2. These end effects loses significance as the number of layers becomes large.
Another way in which this factor is not quite correct is because for a cylinder with inner and outer plates of radius $r$ and $r+d$ and length $L$ the capacitance is not $\frac{\epsilon_0.2\pi rL}{d}$ but $\frac{2\pi\epsilon_0 L}{\ln(1+\frac{d}{r})}$. For large values of $r \gg d$ the latter formula becomes close to the former. This difference is also seen in the fact that when you roll up layers of foil and paper which are exactly the same size when flat, when rolled up the outer layers will not reach as far round as the inner layers. After many turns there can be a significant difference in overlap area.
References :  
Why can rolled up capacitors be modelled as flat parallel plates?
Physics Forums : Rolled up Paper Capacitor
The latter makes reference to S K Foong and C H Lim, "On the capacitance of a rolled capacitor", IOP E-journals, Physics Education, vol 37 no 5, September, 2002 which is accessible via Academia.edu and ResearchGate.net.
