(I) The variable capacitor in the tuner of an AM radio has a...

(I) The variable capacitor in the tuner of an AM radio has a capacitance of $2200 \mathrm{pF}$ when the radio is tuned to a station at $550 \mathrm{kHz}$. What must the capacitance be for a station near the other end of the dial, $1610 \mathrm{kHz?}$

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Key Concepts

Variable Capacitor

A variable capacitor is a component that enables the adjustment of capacitance in an LC circuit. In radio tuners, varying the capacitance allows for fine-tuning the resonant frequency of the circuit to match the frequency of a desired radio station, making it an essential element in selective signal reception.

Resonant Circuit

A resonant circuit, often comprising an inductor and a capacitor, oscillates at a specific frequency determined by the circuit's inductance and capacitance. This natural oscillation, or resonance, is exploited in tuning circuits to select particular frequencies, as the circuit resonates most strongly at its natural frequency.

Frequency-Capacitance Relationship

The resonant frequency of an LC circuit is given by the formula f = 1/(2??(LC)), which shows that the frequency is inversely proportional to the square root of the capacitance. This relationship means that as the capacitance decreases, the resonant frequency increases, and vice versa, a key principle in frequency tuning in radios.

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