Solution 2.9.6.9 R i(t) e(t) C v C (0) Figure 1: Parallel RC circuit For the circuit in Figure 1 applying Kircho 's currentlawyields i(0 + )=i R (0 + )+i C (0 + ): The current through the resistor at time t =0 + is simply i(0 + )=i(t)= v(t) R = V R : At time t =0 + the voltage across the capacitor goes instantaneously from v c (0) to v c (0 + )=V . Then wemust have V ;v c (0) = 1 C Z 0 + 0 i C (t)dt: Unless v c (0) = V ,the left hand side of this equation is a nite number. Since the interval of integration is of zero length, i(t)must be impulsive. That is, we inject nite charge into the capacitor in zero time, requiring i(t) to be a dirac delta function. Thus wehave V ;v c (0) = 1 C Z 0 + 0 K(t)dt = K C : Thus, K =[V ;v c (0)]C;; and i c (t)=[V ;v c (0)]C(t): Finally, i(t)=i R (t)+i C (t)= V R +[V ;v c (0)]C(t): 1