Chapter 10, Problem 12P

The circuit shown in Fig. P10.12 consists of a resistor and capacitor in parallel that are subjected to a $constant$ current source $I$. At time t = 0, the initial voltage across the capacitor is zero. For time $t\ge 0$. the voltage across the capacitor satisfies the following first-order differential equation and initial condition:

$C\frac{dv\left(t\right)}{dt}+\frac{v\left(t\right)}{R}=I$.

(a) Find the transient solution, $v_{tran}\left(t\right)$.

(b) Find the steady-state solution $v_{ss}\left(t\right)$.

(c) Determine the total solution for $v\left(t\right)$ if $v\left(0\right)=0\text{V}$.

(d) Calculate the voltage at times $t=RC,2RC,4RC$, and $t\to \infty$. Use your results to sketch $v\left(t\right)$.