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The quality factor is a dimensionless number which quantifies how efficiently a system stores energy and how strongly it responds on resonance. For a circuit consisting of a capacitor $C$, an inductor $L$, and a small resistance $R$ in series, the resonant frequency is approximately $\omega_0 = 1 / \sqrt{LC}$, and the quality factor, assumed to be large throughout this problem, is

\[
Q = \frac{1}{R} \sqrt{\frac{L}{C}}.
\]

In this problem, we explore several ways to measure $Q$. Uncertainty analysis is not required.

\begin{enumerate}
    \item[(a)] Alice measures $Q$ by seeing how oscillations in the circuit damp over time. Suppose that initially, the charge on the capacitor is $q$ and the current is zero. The next time the current is zero, the charge is $-q(1 - \delta)$. Find an approximate expression for $\delta$, in terms of $\omega_0$ and $Q$.

    \item[(b)] Bob and Charles drive their circuits with a sinusoidal voltage $V(t) = V_0 \cos{\omega t}$. It can be shown that in the steady state, the voltage across the capacitor oscillates with amplitude

    \[
    V_C = \frac{V_0}{\sqrt{(1 - \omega^2/\omega_0^2)^2 + (\omega/\omega_0 Q)^2}}.
    \]

    The circuits Bob and Charles have are similar, but are not precisely the same.
    \begin{enumerate}
        \item[(i)] Bob fixes the value of $V_0$ so that the highest value of $V_C$ at any frequency is precisely $10.00 \, \text{V}$. His equipment can precisely compare the amplitudes of a small DC and AC voltage. He thus performs two very accurate voltage measurements.

        \[
        \begin{array}{|c|c|}
        \hline
        \omega \, (\text{rad/s}) & V_C \, (\text{Volts}) \\
        \hline
        0.0 & 0.1219 \\
        183.3 & 0.1219 \\
        \hline
        \end{array}
        \]

        Using this data, find the numeric values of $Q$ and $\omega_0$ as accurately as possible.

        \item[(ii)] Charles can precisely tune $\omega$, but cannot precisely measure small voltages. He thus fixes $V_0$ to some other value and takes data near the resonance, where $V_C$ is relatively large.

        \[
        \begin{array}{|c|c|}
        \hline
        \omega \, (\text{rad/s}) & V_C \, (\text{Volts}) \\
        \hline
        133.0 & 3.64 \\
        133.5 & 4.76 \\
        134.0 & 6.52 \\
        134.5 & 8.53 \\
        135.0 & 8.18 \\
        135.5 & 6.06 \\
        136.0 & 4.44 \\
        136.5 & 3.42 \\
        137.0 & 2.75 \\
        \hline
        \end{array}
        \]

        Using this data, find the numeric values of $Q$ and $\omega_0$ as accurately as possible. (Hint: you may use the graph paper in the answer sheets, but full credit is attainable without graphing. To find $Q$, you should first find $\omega_0$, then simplify the equation above using $\omega \approx \omega_0$.)

    \end{enumerate}

    \item[(c)] The gain function of this circuit is defined as $G = V_R / V_0$, where $V_R$ is the amplitude of the voltage across the resistor, as shown below.

    \begin{center}
        \includegraphics[width=0.5\textwidth]{circuit_diagram.png}
    \end{center}

    \begin{enumerate}
        \item Find an expression for $G$ in terms of $\omega$, $\omega_0$, and $Q$.
        \item This setup can be used to reject voltages at certain frequencies. Qualitatively describe the range(s) of frequencies for which $G$ is small.
    \end{enumerate}

\end{enumerate}

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