(Solved): 2. Recall the RLC series circuit equation: Ldl2d2Q+RdldQ+C1Q=E. (a) Let z represent the cu ...

2. Recall the RLC series circuit equation: $L\frac{d^{2}Q}{d{l}^2}+R\frac{dQ}{dl}+\frac{1}{C}Q=E$. (a) Let $z$ represent the current of the system $\left(z=\frac{dQ}{dl}\right)$. Assume that $Q(0)=Q_0$. Show that $z$ has to be a solution of the following integro-differential equation, $L\frac{dz}{dt}+Rz+\frac{1}{C}\left[Q_0+{?}_0^{t}z(?)d?\right]=E.$ (b) Let $R=110?,L=1\mathrm{H}$, and $C=0.001\mathrm{F}$. At time $t=0$ there is no charge in the capacitor and zero current. Suppose that the voltage source is turned on and provides $E=90\mathrm{V}$ from time $t=0$ until $t=1$ after which it is turned off again $(E=0$ for $t>1)$. Using Laplace transforms and the results of Problem 1, find $z(t)$. (c) Use the RLC values from part (b) and take the Laplace transform of the following second order system, $L\frac{d^{2}z}{dt}+R\frac{dz}{dt}+\frac{1}{C}z=?90?(t?1).$ What should you assume for the values of $z(0)$ and $z^{?}(0)$ so that your answer is the same as that in part (b)?