(Solved): 4. Charging a capacitor. Consider the RC circuit shown in Fig. 1. Figure 1: RC circuit model. (a) V ...

4. Charging a capacitor. Consider the RC circuit shown in Fig. 1. Figure 1: RC circuit model. (a) Verify that the the differential equation relating the source voltage $v_s$ to the voltage drop across the capacitor $v_C$ is given by: $RC{\dot{v}}_C+v_C=v_s$ (b) Suppose the capacitor is initially discharged $\left(v_C(0)=0\right)$, and we connect a 5 Volt battery to $v_s$. Find $v_C(t)$ by solving the differential equation above using the method of Laplace transforms, and explain why ${\lim }_{t\to \infty }{v}_C(t)=5$. (c) Now that the capacitor is fully charged $\left(v_C(0)=5\right)$, we disconnect the battery. Find $v_C(t)$ under these new conditions, and explain why ${\lim }_{t\to \infty }{v}_C(t)=0$. Note: Such circuits (also called low-pass filters) are used to smooth out noisy signals. For example, if we want to power a sensitive circuit but our source $v_s$ is unreliable and produces randomly fluctuating voltage, we can connect our circuit to $v_C$ instead of connecting it to $v_s$ and we will have a smoother more reliable voltage.