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 = 14.0 \, \Omega[/tex], and the inductance is [tex]L = 1.60 \, \mu \text{H}[/tex]. What capacitance (in [tex]\text{pF}[/tex]) should be used?

To tune into an FM station broadcasting at 99.7 MHz using an RLC circuit with an inductance of 1.60 pH, a capacitance of approximately 2.52 pF should be used.

Explanation:

The resonant frequency of an RLC circuit (composed of resistance, inductance, and capacitance) which is used in a radio to tune into a specific FM station, is given by the formula: f = 1/(2π√LC), where f is the frequency, L is the inductance, and C is the capacitance.

Given the frequency of the radio station f = 99.7 MHz = 99.7 x 10^6 Hz, and the inductance L = 1.60 pH = 1.60 x 10^-12 H, we can solve this equation for C (capacitance). Rearranging the formula gives us C = 1 / (4π²f²L). Plugging the values into this equation, we obtain C ≈ 2.52 pF.

Therefore, a capacitance of approximately 2.52 pF should be used in the RLC circuit to tune into the given FM station.

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