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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