An inductor is connected across a capacitor whose capacitance can be varied by turning a knob. We wish to make the frequency of oscillation of this LC circuit vary linearly with the angle of rotation of the knob, going from$\mathbf{2}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathbf{Hzto}\mathbf{}\mathbf{}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{5}}\mathbf{}\mathbf{Hz}$as the knob turns through 180°. If $\mathit{L}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{mH}$, plot the required capacitance C as a function of the angle of rotation of the knob.

1. Value of the inductance of the inductor,$L=1.0\mathrm{mH}\mathrm{or}1.0\times {10}^{-3}\mathrm{H}$
2. Frequency of the oscillation linearly varies from $f_0=2\times {10}^5\mathrm{Hz}$to as the knob turn through $\theta ={180}^0$.

By using the linear relationship between the angle and frequency, we can find the capacitance in Pico-farads for various angles and plot the capacitance as a function of angle.

Formulae:

The frequency of an oscillation due to the rotation of the knob, $f=f_0\left(1+\frac{\theta }{180°}\right)\left(\mathrm{i}\right)$

The frequency of an oscillation, $f=\frac{\omega }{2\pi }\left(\mathrm{ii}\right)$

The angular frequency of an LC oscillation,$\omega =1/\sqrt{LC}\left(\mathrm{iii}\right)$

Substituting value of angular frequency from equation (iii) in equation (ii), we can get the frequency of the oscillations as follows:

$f=\frac{1}{2\pi \sqrt{LC}}$

Rearrange this formula for the capacitance and substituting frequency value from equation (i), we get

$\begin{array}{c}{f}^2=\frac{1}{4{\pi }^{2}LC}\\ C=\frac{1}{4{\pi }^{2}L{f}^2}\\ =\frac{1}{4{\pi }^{2}L{f}_0^2{\left(1+\frac{\theta }{180°}\right)}^2}\end{array}$

By putting the value of the inductance and the inductance and the given value of lower frequency$f_0$, we get the capacitance as

$\begin{array}{c}C=\frac{{180}^2}{4{\pi }^2\left(1.0\times {10}^{-3}\right)\left(2\times {10}^5\right){\left(180+\theta \right)}^2}\\ =\frac{{180}^2}{8{\pi }^2{\left(180+\theta \right)}^2}\times {10}^{10}\mathrm{pF}\end{array}$

Now vary the θ and measure the capacitance in the Pico farad shown in the following table:

Now plot the value $C\left(\mathrm{pF}\right)\mathrm{versus}\theta$:

By using capacitance as a function of angle, we plotted on the graph.