Question

LCoscillators have been used in circuits connected to loudspeakers to create some of the sounds of electronic music. What inductance must be used with a 6.7 µF capacitor to produce a frequency of10 kHz, which is near the middle of the audible range of frequencies?

Short answer

The inductance to be used is$3.78\times {10}^{-5}\text{H}$ , which is near the middle of the audible range of frequencies.

Step-by-step solution

The given data

1. The capacitance of the capacitor,$C=6.7\mathrm{\mu F}$.
2. The frequency that the capacitor will produce$f=10\text{kHz or}10000\text{Hz}$.

Understanding the concept of inductance

Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. By using the direct relationship between the frequency of inductance and the inductance, we can find the inductance.

Formulae:

The resonance frequency of an LC oscillation, $f=\frac{1}{2\pi \sqrt{LC}}$. (i)

Calculation of the value of inductance

By using equation (i), we can find the inductance to be used having frequency near the middle of the audible range of frequencies as follows:

$\begin{array}{rcl}L& =& \frac{1}{{\left(2\pi \right)}^2{f}^{2}C}\\ & =& \frac{1}{{\left(2\pi \right)}^2{10000}^2\times 6.7\times {10}^{-6}}\\ & =& \frac{1}{{\left(2\pi \right)}^2{10000}^2\times 6.7\times {10}^{-6}}\\ & =& 3.78\times {10}^{-5}\text{H}\\ & & \end{array}$

Hence, the value of the inductance is $3.78\times {10}^{-5}\text{H}$.