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An LC circuit is a fundamental component in many electronic devices, including car alarms. It consists of an inductor (L) and a capacitor (C) connected in series or parallel. The beauty of an LC circuit lies in its ability to oscillate current at a particular frequency, known as the resonant frequency. This resonance is crucial for applications like generating sound at a specific frequency, as in the car alarm example.

The resonant frequency of an LC circuit is determined by the formula:

• \[ f = \frac{1}{2\pi \sqrt{LC}} \]

This formula establishes a link between the inductance (L) and capacitance (C), dictating the frequency at which the circuit oscillates. At this frequency, energy transfers back and forth between the inductor and the capacitor, resulting in a sustained oscillation. This principle allows your car alarm to emit a tone at the desired frequency, enhancing its effectiveness in alerting of unauthorized access.

Capacitors are electronic components that store and release electrical energy. In the context of the LC circuit for a car alarm, the capacitor must hold a specific amount of energy to produce adequate sound. This energy storage capability is described by the formula for the energy (\(U\)) stored in a capacitor.

• \[ U = \frac{1}{2} C V^2 \]

Here, \(U\) is the energy stored, \(C\) is the capacitance, and \(V\) is the voltage across the capacitor. For the car alarm to produce a loud enough sound, it needs to store 0.0160 J of energy. Setting this condition allows us to solve for the value of capacitance required. By applying the formula, one can adjust the capacitor's specifications to store the needed energy for the car alarm's operation. This concept is pivotal in ensuring the alarm is not just heard but is also effective in its purpose.

Determining the correct values of inductance (L) and capacitance (C) is a critical step in designing an LC circuit, such as a car alarm. First, using the resonant frequency equation,\(LC = \frac{1}{(2\pi f)^2}\), we derived a relationship between these two variables. This allows us to establish a foundation for further calculations.

Once the capacitance (\(C\)) is calculated using the capacitor energy formula (where \(C \approx 0.222 \mu F\)), the inductance (\(L\)) can then be determined by substituting this value back into the resonant frequency relationship. This calculation gives us the inductance required to achieve resonance at the desired frequency of 3500 Hz, which is approximately \(0.165 H\).

This step-wise approach ensures the car alarm circuit is finely tuned to produce sound at the correct frequency. It also highlights the interdependency between inductance and capacitance, where altering one will affect the necessary value of the other to maintain the system's performance.