Radios use resonance in order to tune in to a particular station. A physics student builds a simple radio using an RLC series circuit. They decide to use a resistor with [tex]R = 43.0 \, \Omega[/tex], but they only have one capacitor with capacitance [tex]C = 210 \, \text{pF}[/tex].

To listen to their favorite station, KXY 97.9 FM, which is at a frequency of [tex]97.9 \, \text{MHz}[/tex], what must be the inductance [tex]L[/tex] of their circuit's inductor?

To determine the required inductance (L) for the circuit, we can use the formula for resonance in an RLC series circuit. Resonance occurs when the inductive reactance (XL) equals the capacitive reactance (XC), which is given by:

XL = 2πfL,
XC = 1/(2πfC),

where f is the frequency of the desired station, given as 97.9MHz or 97.9×10^6 Hz, and C is the capacitance of the capacitor, given as 210pF or 210×10^(-12) F.

Since XL = XC at resonance, we can equate the two equations above and solve for L:

2πfL = 1/(2πfC),
L = 1/(4π^2f^2C).

Plugging in the values, we have:
L = 1/(4π^2(97.9×10^6)^2(210×10^(-12))),
L ≈ 5.19×10^(-7) H.

Therefore, the inductance of the circuit's inductor should be approximately 5.19×10^(-7) H to tune in to the favorite station KXY 97.9FM.

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