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------------------------------------------------ An RLC circuit is used in a radio to tune into an FM station broadcasting at [tex]f = 99.7 \, \text{MHz}[/tex]. The resistance in the circuit is [tex]R = 12.8 \, \Omega[/tex], and the inductance is [tex]L = 1.58 \, \mu\text{H}[/tex]. What capacitance should be used?

To tune an RLC circuit in a radio to a specific frequency, you must adjust its resonant frequency by altering its capacitance. The formula for the resonant frequency of an RLC circuit is used to determine this capacitance based on the given inductance and frequency.

Explanation:

The subject of this question is a Resistor-Inductor-Capacitor (RLC) circuit, in the context of a radio tuning system. The variable component here is the capacitance which needs to be calculated.

Firstly, it's important to understand that the radio is tuned to the desired frequency by adjusting the resonant frequency of its circuitry to match the frequency of the broadcasting station. To find this, we need to make use of the formula that defines the resonant (or natural) frequency of an RLC circuit: f0 = 1 / (2π√LC), where f0 represents frequency, L stands for inductance, and C for capacitance.

Going by the provided info, we can rearrange this formula to solve for C: C = 1 / ((2πf0)²L). Substituting the given values for f0 (99.7x106 Hz or 99.7 MHz) and L (1.58x10-6 H or 1.58 µH), we can find the required capacitance for the RLC circuit to tune into the specified frequency.

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