A variable air capacitor used in a radio tuning circuit is made of N semicircula

Question

A variable air capacitor used in a radio tuning circuit is made of N semicircular plates each of radius R and positioned a distance d from its neighbors, to which it is electrically connected. As shown in the figure below, a second identical set of plates is enmeshed with the first set. Each plate in the second set is halfway between two plates of the first set. The second set can rotate as a unit. Determine the capacitance as a function of the angle of rotation theta, where theta = 0 corresponds to the maximum capacitance. C =

Explanation / Answer

The capacitance of "N" intermeshing plates as described is a group of (2N-1) parallel plate capacitors in parallel.
From the description it sounds like the plate separation between intermeshing plates is d/2.
When the plates have overlapping areas "A" the total capacitance of the group is then;

C = (2N-1)e(o)A/(d/2)

The overlapping areas of semicircle sections are pie shaped and will have areas;

A = (1/2)R^2(pi - theta)

So when theta = 0 you have overlapping full semicircles and when theta = pi you have zero overlapping areas.

C = (2N-1)e(o)(R^2/d)(pi - theta)

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