$($adapted from Prob. $6,$ p$.654,$ Brown's Introductory Physics II$)$ Consider the diagram below. The AC current through both the resistor and capacitor is $i(t)=I_{\text{m}\text{a}\text{x}}sin(t).$

$($a$)$ Find an expression for the voltage $V_s(t)$ that the source must supply in terms of the frequency $$ of the source, the impedance $Z$ of the circuit, the phase shift $$ between the applied voltage and the current, and the amplitude $I_{\text{m}\text{a}\text{x}}$ of the current. Also find expressions for $Z$ and $$ in terms of $R,C,$ and $.$

$($b$)$ Rewrite your expression for $V_s(t)$ so that it is only in terms of $I_0,R,$ and the dimensionless quantities $t,,$ and $RC.$

FYI: The argument to a trig function or exponential should always be dimensionless $($i$.$e$.$ its units must cancel$),$ and this applies to the $t$ that goes inside the sine since $$ is in $s^{-1}$ and $t$ is in s $.$ Angles in radians are by definition dimensionless. The last quantity $(RC)$ is a measure of the ratio of the time constant of the RC circuit to the period of the source driving the circuit, i$.$e$.RC=2(\frac{{}_{RC}}{T_{\text{d}\text{r}\text{i}\text{v}\text{i}\text{n}\text{g}}}).$

$($c$)$ What frequency $$ would minimize 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? Explain this in terms of the voltage response of a capacitor at high and low frequency.