$($adapted from Prob. $4,$ p$.653,$ Brown's Introductory Physics II$)$ Consider the diagram below showing a series RLC circuit. The variable AC source supplies a current $i(t)=I_{\text{m}\text{a}\text{x}}sin(t).$

$($a$)$ Write out expressions for the voltage supplied by the source, the voltage across each circuit element, the phase shift, and the impedance. $($Don$\text{'}$t rederive them, just find them from your notes.$)$ The expressions for the voltages should be written in terms of $I_{\text{m}\text{a}\text{x}},,t,$ and the phase shift and impedance. The expressions for the phase shift and impedance should be written in terms of $R,L,C,$ and $.$

$($b$)$ Find $($and simplify$)$ the average power $($:$P$:$)=\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{}}}P(t)dt=\frac{1}{T}{\displaystyle \underset{0}{\overset{T}{}}}v(t)i(t)dt,$ of each element $(R,L,C,$ and the source, for AC source period $T.$ Which elements dissipate power on average? Note: we did this in class for the resistor and the inductor. Apply the same reasoning to the capacitor.

$($c$)$ Verify that at $t=0,$ the voltage of the source is equal to the sum of the voltages across the resistor, inductor, and capacitor.

$($d$)$ What frequency $($in terms of $R,L,$ and$/$or $C)$ minimizes the impedance of the circuit, thereby requiring the smallest amplitude of voltage $V_{S,\text{m}\text{a}\text{x}}$ to drive a given amplitude $I_{\text{m}\text{a}\text{x}}$ of current?

$($e$)$ The circuit is now operated specifically at the frequency you found in $($d$),$ called the resonant frequency of the circuit. At the time when the instantaneous current through the circuit has its maximum value, find expressions for the voltages across the resistor, the inductor, the capacitor, and the source. After simplification, your expressions should contain only $I_{\text{m}\text{a}\text{x}}$ and $R.$