The FM radio band covers the frequency range \(88-108\) MHz. If the variable capacitor in an FM receiver ranges from \(10.9 \mathrm{pF}\) to \(16.4 \mathrm{pF},\) what inductor should be used to make an \(L C\) circuit whose resonant frequency spans the FM band?

The FM (Frequency Modulation) radio band is a range of electromagnetic frequencies set aside for broadcasting FM radio stations. It falls within the VHF (Very High Frequencies) spectrum, typically between 88 to 108 MHz (Megahertz) in most countries. This band is used to transmit high-fidelity sound over the airwaves.

In our exercise, an FM receiver needs to be tuned to pick up signals across the entire FM band. To achieve this, the receiver's circuitry must be capable of resonating at frequencies as low as 88 MHz and as high as 108 MHz. The choice of components in this circuit, particularly the inductor used with a variable capacitor, will determine the range of receivable frequencies.

An LC circuit, also known as a resonant circuit or tank circuit, is an electrical circuit consisting of an inductor, represented by the symbol 'L', and a capacitor, represented by the symbol 'C'. These components are typically connected in series or parallel.

The LC circuit is fundamental in electronics because it can resonate at a specific frequency, known as the resonant frequency. This property makes it ideal for tuning a circuit to a specific frequency, like those in an FM radio receiver. This type of circuit can filter out unwanted frequencies while allowing the desired frequency range to pass through, which is crucial for the clarity of the received signals.

Inductor-capacitor resonance in an LC circuit occurs when the inductive reactance and capacitive reactance are equal in magnitude but opposite in phase, causing them to cancel each other out. When this happens, the circuit resonates at a particular frequency, which is determined by the values of the inductor and capacitor.

The resonance frequency can be calculated using the formula \( f = \frac{1}{2\pi\sqrt{LC}} \), where 'f' is the frequency in hertz (Hz), 'L' is the inductance in henries (H), and 'C' is the capacitance in farads (F). Resonance is significant for our FM radio band exercise because selecting the correct value of the inductor will allow the LC circuit to resonate and thus correctly receive the signals across the specified FM frequency range.

Electromagnetic frequencies refer to the oscillations per second of electromagnetic waves. These frequencies form the electromagnetic spectrum, which ranges from very low frequencies, like those used in power lines, to extremely high frequencies, like gamma rays. Each frequency has its own set of properties and uses in different technologies.

FM radio band frequencies lie in the VHF part of the spectrum and are particularly suitable for broadcasting because they can deliver a clear sound over long distances and have a good resistance to noise and interference. Understanding how these various electromagnetic frequencies work is essential when designing circuits that need to receive or transmit a specific range of frequencies, such as the LC circuit for an FM radio receiver.