A radio can be tuned into a particular station frequency by adjusting the capaci

Question

A radio can be tuned into a particular station frequency by adjusting the capacitance in an L-Ccircuit. Suppose that the minimum capacitance of a variable capacitor in a radio is 4.12 pF .

Part A )What is the inductance L of a coil connected to this capacitor if the oscillation frequency of the L-C circuit is 1.65 MHz , corresponding to one end of the AM radio broadcast band, when the capacitor is set to its minimum capacitance? Express answer in henrys to three significant figures.

Part B) The frequency at the other end of the broadcast band is 0.547 MHz . What is the maximum capacitance Cmax of the capacitor if the oscillation frequency is adjustable over the range of the broadcast band? Express answer in farads to three significant figures.

Explanation / Answer

The natrural frequency of an LC circuit is
f = 1/( 2· ·(L·C) )

So:
f_max = 1/( 2· ·(L·C_min) )
<=>
L = 1/( (2··f_max)² ·C_min)
= 1/( (2· · 1.65×10Hz)² · 4.12×10¹² F)
= 2.34×10³H
= 2.34×10mH

and

f_min = 1/( 2· ·(L·C_max) )
<=>
C_max = 1/( (2··f_max)² ·L)
= 1/( (2· · 0.547×10Hz)² · 2.34×10³H)
= 3.617×10¹¹F
= 36.1pF

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