Question

A radio can be tuned into a particular station frequency byadjusting the capacitance in an L-C circuit. Suppose thatthe minimum capacitance of a variablecapacitor in a radio is4.14 .a) What is the inductance of a coil connected to this capacitor ifthe oscillation frequencyof the L-C circuit is 1.70 , corresponding to one end of the AM radio broadcast band,when the capacitor is set to itsminimum capacitance?b) The frequency at the other end of the broadcast band is0.538 . What is the maximum capacitance $C_max$ of the capacitor if the oscillation frequency isadjustable over the range of the broadcastband?

Answer 1

Hi,
Given data:
The minimum capacitance of a variable capacitor in a radio = 4.14pF =4.14x10^-12F.
( a ) Frequency of the L-C circuit = 1.70MHz = 1.7x10⁶
Findinductance L ?

In this case, if we want to tune any frequency ( i.eresonant frequency ) conditionis,
X_c =X_L
∴ 1 / 2πfC =2πfL
1 / ( 2 x3.14 x 1.7 x 10⁶ x 4.14 x 10^-12 ) = 2 x3.14 x 1.7 x 10⁶ x L

Simplifyand get L value = 1.48 mH


( b ) The other end of the broadcast band is0.538 MHz
FindC_max = ?
L = 1.48mH ( Taken from above problem (Fixed))

X_c = X_L ; 1 /2πfC = 2πfL

C= 1 / ( 4 π²f² L )
C_max = 1 / ( 4π² x (0.538 x 10⁶ )²x 1.48 x 10^-3 )
C_max =0.237 x 10^-9 = 0.237 nF

I hope this will helps you.