Respuesta :
Complete Question
The FM radio band covers the frequency range 88-108 MHz
If the variable capacitor in an FM receiver ranges from 14.9 pF to 22.4 pF , what inductor should be used to make an LC circuit whose resonant frequency spans the FM band
Answer:
The inductor that should be made use of is and inductor with an inductance of [tex]L = 1.46 *10^{-7} \ H[/tex]
Explanation:
From the question we are told that
The frequency range for the FM band is 88-108 MHz
The range of the capacitance is [tex]C = 14.9 \ pF \ \ to \ 22.4 pF = 14.9 *10^{-12} \ F \ \ to \ \ 22.4 *10^{-12}\ F[/tex]
The resonant frequency is [tex]f_r = 108 MHz. = 108 *10^6 \ Hz[/tex] this is in the question we are told that the resonant frequency spans the FM band
Generally the at resonance the reactance of the capacitor is equal to that of the inductor i.e
[tex]R_c = R_l[/tex]
Here [tex]R_c[/tex] is the reactance of the capacitor which is mathematically represented as
[tex]R_c = \frac{1}{2 \pi f C}[/tex]
while the reactance of the inductor is [tex]R_i = 2 \pi f L[/tex]
So
[tex]\frac{1}{2 \pi f C} = 2 \pi f L[/tex]
=> [tex]L = \frac{1}{4 \pi^2 * f ^2 * C }[/tex]
When C is [tex]C = 14.9 \ pF = 14.9 *10^{-12} \F[/tex]
=> [tex]L = \frac{1}{4 (3.142 )^2 * (108*10^6) ^2 * 14.9 *10^{-12} }[/tex]
=> [tex]L = 1.46 *10^{-7} \ H[/tex]
When C is [tex]C = 22.4 \ pF = 14.9 *10^{-12} \F[/tex]
=> [tex]L = \frac{1}{4 (3.142 )^2 * (108*10^6) ^2 * 22.4 *10^{-12} }[/tex]
=> [tex]L = 1.46 *10^{-7} \ H[/tex]