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Question

A variable air capacitor in a tuning circuit is made up of N semicircular plates, each of radius R and positioned at a distance d from its neighbours, to which it is electrically connected as shown in figure, a second identical set of plates is enmeshed with its plates halfway between those of the first set. The second set can rotate as a unit. Determine the capacitance as a function of the angle of rotation θ, where θ=0 corresponds to the maximum capacitance.


A
2Nsinθ πR2ε0d
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B
(2N1)sinθ πR2ε0d
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C
2N(πθ)R2ε0d
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D
(2N1)(πθ)R2ε0d
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Solution

The correct option is D (2N1)(πθ)R2ε0d
When θ=180 or π; the plates are out of mesh and the overlap area is zero.

With θ=0; the overlap area between the adjacent plates is that of a semicircle, i.e, πR22

The effective area of a capacitor formed by a pair of adjecent plates as function of θ radian is,

A=(πθ)R22

As there are two identical coils each having N sets of plates,

Total no. of plates =2N

In a pair of plates, distance between them =d2

Capacity of one capacitor formed is,

C=Aε0d=((πθ)R22)ε0d2

C=(πθ)R2ε0d

For total 2N plates connected alternately, it will form (2N1) capacitors, all are in parallel combination.

Ceff=(2N1)(πθ)R2ε0d

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, (D) is the correct answer.

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