A broadcasting station transmits at a frequency of 89.5 MHz. What capacitance together with an inductance of \(30 \mu H\) is needed to tune to this station?
A. \(4.16 nF \) B. \(2.68 KF\) C. \(4.16 pF\) D. \(2.98 \mu F\)
Correct Answer:
Explanation
To tune to this station, the capacitance and inductance must be at resonancet resonance, \(X _{ L }= X _{ C }\) $$ \begin{aligned} &\frac{1}{2 \pi f C}=2 \pi f L \\ &\text { So that } f=\frac{1}{2 \pi \sqrt{L C}} \text { Or } \\ &C=\frac{1}{4 \pi^{2} f^{2} L} \\ &=\frac{1}{\left[4 \times(3.142)^{2}\left(89.5 \times 10^{6}\right)^{2}\right]} \\ &C=\frac{1}{4 \times 9.872 \times 8.028 \times 10^{-6} \times 30 \times 10^{-6}} \\ &=\frac{1}{9510.63710^{9}}=\frac{1}{9.51 \times 10^{12}} \\ &=1.05 \times 10^{-13} F \\ &=10.5 \times 10^{-12} F =10.5 PF \end{aligned} $$
