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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 = 11.4 \, \Omega[/tex], and the inductance is [tex]L = 1.42 \, \text{H}[/tex]. What capacitance should be used?

____________ pF

To tune into a FM radio broadcasting at 99.7 MHz using an RLC circuit with a resistance of 11.4 ohms and an inductance of 1.42 H, the required capacitance would be approximately 1.27 picofarads (pF). This is calculated by applying the formula for frequency in an RLC circuit.

Explanation:

In an RLC circuit, the frequency of a radio signal is related to the inductance (L) and the capacitance (C) in the circuit by the formula: f = 1 / (2π √(LC)). You are asked to find the needed capacitance to tune into a FM station broadcasting at 99.7 MHz.

Given the frequency (f) is 99.7 MHz which, converted to Hz, is 99.7 x 10^6 Hz and the inductance (L) is 1.42 H, we can plug these values into the formula and solve for C:

C = 1 / (4π^2 f^2 L) = 1 / (4π^2 *(99.7*10^6)^2 * 1.42) = 1.27 x 10^-12 F. Convert this value to pF by multiplying by 10^12, so the appropriate capacitance needed for this radio to tune into the FM station is approximately 1.27 pF.

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