Electronic – Miller’s Theorem – Input Capacitance

The total capacitance can be calculated in just a few steps using the basic rules for combining capcitors in series and parallel.

To get you started, three 6 uF capcitors in series is equivalent to a 2 uF capacitor.

And a 2 uF capacitor in parallel with a 4 uF capacitor is equivalent to a 6 uF capacitor.

\$\epsilon_r\$ of 2 is kind of small for plastics, a lot are more like 3 or 4, but that's not too important.

As to why it (sort-of) doubles- think about the physical arrangement. The second sheet of plastic (the one that is not between the metal sheets) is doing nothing before you roll it up. When you roll it up, into (say) n turns, you will be using both sides of each sheet of metal, except for the last turn inside and the first turn outside. So, if n is big enough, it approximately doubles (if the roll starts at zero radius, the inside turn won't matter).

You should be able to estimate the radius easily from the standard mensuration formulas for solids. The rectangular prism will have volume of \$V_p = 2t\cdot L\cdot W\$ where t is the thickness of a single sheet of plastic. The cylinder will have a volume of \$V_c = \pi r^2 \cdot L_C\$ where \$L_C\$ is the length of the cylinder. So, depending on how you roll (so to speak), \$L_C\$ will be either L or W from the sheet and by equating the volumes, you can solve for r (which assumes that the inner radius is zero).