Variable capacitor question

[pc2-brazil]

Homework Statement

The figure (attached) shows a variable capacitor which uses air as the dielectric, of the kind used in the syntonization of radio devices. The plates are connected alternately; a group of them is fixed, while the other group can rotate around an axis. Consider a set of n plates of alternated polarity, each one of them having an area A and separated by the distance d. Show that the maximum value of the capacitance is:
$$C = \frac{(n-1)\epsilon_0 A}{d}$$

2. The attempt at a solution
It seems that there are (n - 1) plates in series, but this wouldn't lead to the expected result. According to the expected result, the capacitors would have to be in parallel, then, the maximum capacitance would occur when the movable plates aren't rotated with respect to the fixed plates. But it doesn't seem that they are in parallel. Could someone explain?

Thank you in advance.

 

[gneill]

Why do you think that the plates are in series? Can you draw a circuit diagram that corresponds to the scenario?

 

[pc2-brazil]

Why do you think that the plates are in series? Can you draw a circuit diagram that corresponds to the scenario?

The plates appear to be in series because each capacitor shares a plate with the adjacent one.

 

[gneill]

The plates appear to be in series because each capacitor shares a plate with the adjacent one.

But the plates are connected externally. There are two sets of plates that are interleaved. Each set is connected so that they share the same potential. Each plate "opposes" one or two plates of the opposite set. Each pair of 'opposing' plate surfaces represents a separate capacitor.

 

[rwisz]

I would like to offer some clarification on the concept of parallel and series capacitors in this scenario. In this variable capacitor, the plates are connected alternately, meaning that a fixed plate is followed by a rotating plate, and so on. This creates a series of parallel plate capacitors, where each capacitor has a fixed plate and a rotating plate.

To find the maximum capacitance, we need to consider the configuration where all the rotating plates are aligned with the fixed plates (i.e. not rotated). In this scenario, the distance between each fixed plate and its corresponding rotating plate is the same, and the plates are all parallel to each other. This creates a series of parallel capacitors, with each capacitor having a capacitance of C = ε0A/d, where ε0 is the permittivity of air, A is the area of each plate, and d is the distance between the plates.

Since there are (n-1) pairs of fixed and rotating plates, the total capacitance is given by:
Ctotal = (n-1)C = (n-1)ε0A/d

Therefore, the maximum capacitance in this variable capacitor is:
Cmax = (n-1)ε0A/d

I hope this explanation helps to clarify any confusion. It is important to note that while the plates are connected alternately, they are still considered to be in parallel when determining the maximum capacitance.