High School

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 = 12.0 \, \Omega[/tex], and the inductance is [tex]L = 1.40 \, \mu\text{H}[/tex]. What capacitance should be used?

Answer :

The capacitance that should be used in the RLC circuit is approximately 1.14 x 10^-13 Farads.

To calculate the capacitance needed in the RLC circuit, we can use the formula:

\(f = \frac{1}{2\pi\sqrt{LC}}\)

Given:
Frequency (f) = 99.7 MHz = 99.7 x 10^6 Hz
Resistance (R) = 12.0 Ω
Inductance (L) = 1.40 µH = 1.40 x 10^-6 H

First, convert the frequency to radians per second by multiplying it by 2π:

\(ω = 2πf\)

Substitute the values into the formula:

\(2πf = \frac{1}{\sqrt{LC}}\)

Simplify the formula:

\(LC = \frac{1}{(2πf)^2}\)

Plug in the given values:

\(LC = \frac{1}{(2π(99.7 x 10^6))^2}\)

Solve for LC:

\(LC = 4.02 x 10^{-14}\)

Now, divide both sides of the equation by L to solve for C:

\(C = \frac{1}{L(2πf)^2}\)

Plug in the values:

\(C = \frac{1}{(1.40 x 10^{-6})(2π(99.7 x 10^6))^2}\)

Solve for C:

\(C = 1.14 x 10^{-13}\) Farads

Therefore, the capacitance that should be used in the RLC circuit is approximately 1.14 x 10^-13 Farads.

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