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 $\theta$, where $\theta =0^{\circ }$ corresponds to the maximum capacitance.

• A. $\frac{2N\sin \theta \pi R^2{\epsilon }_0}{d}$
• B. $\frac{(2N-1)\sin \theta \pi R^2{\epsilon }_0}{d}$
• C. $\frac{2N(\pi -\theta )R^2{\epsilon }_0}{d}$
• D. $\frac{(2N-1)(\pi -\theta )R^2{\epsilon }_0}{d}$

Answer 1

The correct option is D $\frac{(2N-1)(\pi -\theta )R^2{\epsilon }_0}{d}$

When $\theta ={180}^{\circ }or\pi$; the plates are out of mesh and the overlap area is zero.

With $\theta =0^{\circ }$; the overlap area between the adjacent plates is that of a semicircle, i.e, $\frac{\pi R^2}{2}$

$\therefore$ The effective area of a capacitor formed by a pair of adjecent plates as function of $\theta \text{radian}$ is,

$A=\frac{(\pi -\theta )R^2}{2}$

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

$\therefore$Total no. of plates $=2N$

In a pair of plates, distance between them $=\frac{d}{2}$

$\therefore$ Capacity of one capacitor formed is,

$C=\frac{A{\epsilon }_0}{d}=\frac{\left(\frac{(\pi -\theta )R^2}{2}\right){\epsilon }_0}{\frac{d}{2}}$

$\therefore C=\frac{(\pi -\theta )R^2{\epsilon }_0}{d}$

For total $2N$ plates connected alternately, it will form $(2N-1)$ capacitors, all are in parallel combination.

$\therefore C_{eff}=\frac{(2N-1)(\pi -\theta )R^2{\epsilon }_0}{d}$

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