An FM radio station broadcasts at a frequency of 100 MHz. What inductance should be paired with a 10 pF capacitor to build a receiver circuit for this station?

We have given that, an FM radio station broadcasts at a frequency of 100 MHz, and receiver circuit for this station has 10 pF capacitor.

The formula to calculate the inductance is :

$L=\frac{1}{{\omega }^{2}C}$

Here, $L$= Inductance

$\omega$= Resonant frequency

$C$= Capacitance

We have to get value of resonant frequency first.

Formula of resonant frequency is :

localid="1650299962559" $\omega =\left(2\mathrm{\pi }\mathrm{f}\right)$

Here, $\omega =$Resonant frequency

f = Frequency at which station broadcast

Substituting the values in the formula of resonant frequency :

localid="1651031527615" $\omega =2\mathrm{\pi }(100\times {10}^{6}Hz)$

localid="1650300275603" $\omega =6.28\times {10}^{8}Hz$

Now we will substitute all the values in the formula of inductance.

Substituting the values in the inductance formula :

$L=\frac{1}{{\omega }^{2}C}$

localid="1651032743574" $L=\frac{1}{{(6.28\times {10}^{8}Hz)}^2(10\times {10}^{-12}F)}$

$L=2.5\times {10}^{-7}H=0.25\mu H$

$0.25\mu H$inductance should be paired with a 10 pF capacitor to build a receiver circuit for this station.