Physics Problem Solving For High School - A series RLC circuit is used in a radio to tune an FM station broadcasting at 103.7 MHz. The resistance in the circuit is 10$\Omega$ and the inductance is 2.0 $\mu H$. What is the best estimate of the capacitance that should be used?
A. 200 pF
B. 50 pF
C. 1.25 pF
D. 0.20 pF
E. 0.02 pF
Physics Solution: C
\[\begin{align*} f &= 103.7 \times 10^{6} Hz\\ &\approx 10^{8} Hz\\\\ L &= 2.0 \mu H\\&= 2 \times 10^{-6} H\\\\ X_L &= X_C\\2\pi f L&=\frac{1}{2\pi f C}\\ C&= \frac{1}{ 4\pi ^2f^2L} \\C&= \frac{1}{ 4\pi^2\times (10^{8})^2(2 \times 10^{-6})} \\ C&\approx \frac{1}{ 8\pi^2\times 10^{10} } \\C&\approx \frac{1}{ 8 } \times 10^{-11}\\C&\approx 1.25 \times 10^{-12}\\C&= 1.25pF\\ \end{align*}\]
