A simple $RC$ circuit can be constructed by connecting a capacitor ($C$), a resistor ($R$), and a battery ($\mathcal{E}$) in series, as shown in Figure 3.1. When the circuit is closed (at $t = 0$), current immediately flows through the circuit, causing electric charges to accumulate on the capacitor plates. As a result, the potential difference across the capacitor ($\Delta v_C$) grows with time, from zero to $\mathcal{E}$, according to the equation \begin{equation} \Delta v_C(t) = \mathcal{E} \left( 1 - e^{-t/RC} \right) \label{lab3:eq:V_charging} \end{equation} Assume that Equation \ref{lab3:eq:V_charging} is still a theory. Perform an experiment, as suggested in the procedure, to investigate the time dependence of the potential difference across a capacitor during the charging process.
When a fully charged capacitor is allowed to discharge (see Figure 3.2) through a resistor $R$, current flows out of the capacitor causing a gradual depletion of the charges on its plates. As a result, the potential difference across the capacitor decays with time, from $\mathcal{E}$ to zero, according to the equation \begin{equation} \Delta v_C(t) = \mathcal{E}\, e^{-t/RC} \label{lab3:eq:V_discharging} \end{equation} Here also, assume that Equation \ref{lab3:eq:V_discharging} is still a theory, and perform suitable experiments to test its validity, as suggested in the procedure.
