Problem 26.58 An electronics hobbyist is building a radio set to receive the AM

Question

Problem 26.58 An electronics hobbyist is building a radio set to receive the AM band, with frequencies from 520 kHz to 1500 kHz . The antenna, which also serves as the inductor in an LC circuit, has an inductance of 251 1iH . She needs to add a variable capacitor whose capacitance she can adjust to tune the radio. Part A What is the minimum capacitance the capacitor must have? Express your answer to two significant figures and include the appropriate units. Part B What is the maximum capacitance the capacitor must have? Express your answer to two significant figures and include the appropriate units.

Explanation / Answer

Given

fmin = 520 kHz

fmax = 1500 kHz

L = 251 micro H

A) we know resonant frquency, f = 1/(2*pi*sqrt(L*C))

when c is minimum f is maximum

so,

fmax = 1/(2*pi*sqrt(L*Cmin))

fmax^2 = 1/(4*pi^2*L*Cmin)

Cmin = 1/(4*pi^2*L*fmax^2)

= 1/(4*pi^2*(251*10^-6)*(1500*10^3)^2)

= 4.48*10^-11 F

B) when c is maximum f is minimum

so,

fmin = 1/(2*pi*sqrt(L*Cmax))

fmin^2 = 1/(4*pi^2*L*Cmax)

Cmax = 1/(4*pi^2*L*fmin^2)

= 1/(4*pi^2*(251*10^-6)*(520*10^3)^2)

= 3.73*10^-10 F

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