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Q100PE

Expert-verifiedFound in: Page 864

Book edition
1st Edition

Author(s)
Paul Peter Urone

Pages
1272 pages

ISBN
9781938168000

**The lowest frequency in the FM radio band is 88.0 MHz ****. (a) What inductance is needed to produce this resonant frequency if it is connected to a 2.50pF ****capacitor? (b) The capacitor is variable, to allow the resonant frequency to be adjusted to as high as 108 MHz****. What must the capacitance be at this frequency?**

- The inductance needed to produce this resonant frequency is $1.31\mu H$.
- The capacitance at this frequency is $1.66pF$.

**A capacitor is a two-terminal electrical component that may store energy in the form of an electric charge. It is made up of two electrical wires separated by a certain distance. The space between the conductors can be filled with the vacuum or a dielectric, which is an insulating substance.**

Remember that in a series circuit with a coil and a capacitor, the resonant frequency is given by

$f=\frac{1}{2\pi \sqrt{LC}}$ …………….(1)

The coil's inductance can be expressed as follows by rearranging the above equation (1), such that,

$L=\frac{1}{4{\pi}^{2}{f}^{2}C}$ …………….(2)

Substituting the given data in equation (2) we can get,

$L=\frac{1}{4{\pi}^{2}\times {\left(8.8\times {10}^{7}Hz\right)}^{2}\times \left(2.50\times {10}^{-12}F\right)}\phantom{\rule{0ex}{0ex}}=1.31\times {10}^{-6}H\left(\frac{1\mu H}{{10}^{-6}H}\right)\phantom{\rule{0ex}{0ex}}=1.31\mu H$

Therefore, the inductance needed to produce this resonant frequency is $1.31\mu H$.

We can solve for the value of the capacitor at a new frequency if we know the coil inductance, using the equation (2) such that,

$C=\frac{1}{4{\pi}^{2}{f}^{2}L}$ ………………(3)

Substituting the given data in the above equation (3), we'll have

$C=\frac{1}{4{\pi}^{2}\times {\left(1.08\times {10}^{8}Hz\right)}^{2}\times \left(1.31\times {10}^{-6}H\right)}\phantom{\rule{0ex}{0ex}}=1.66\times {10}^{-12}F\left(\frac{1pF}{{10}^{-12}F}\right)\phantom{\rule{0ex}{0ex}}=1.66pF$

Therefore,** **the capacitance at this frequency is 1.66 pF .

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