7.4 Charging and Discharging Case

Combining the results of the previous two chapters, we now consider the case in which a capacitor, having an initial voltage Vo, is charged by battery voltage V_B, in the circuit shown.  Rather than developing the differential equation for this case, we provide sketches and the governing formula by combining our previous results. The capacitor voltage is initially Vo, and it exponentially approaches VB with time. Two cases are plotted, one in which V_B>V_o, and the other in which V_B<V_o.

The governing equation for this case is given below. We also show the results of a simple check for this equation, verifying that it predicts v_c(t) equal to V_o at t=0 and V_B when t approaches infinity.