For a RC series circuit (Fig. 19-10), find (a) the time constant of the circuit; (b) v_{C} and v_{R} one time constant after the switch is closed and at 5 s, and (c) v_{C} and v_{R} one time constant after discharge starts, assuming the capacitor is fully charged to 10 V.
(a) Write Eq. (19-23) for the time constant in a RC series circuit and substitute values for R and C.
T = RC (19-23)
=\left(100 \times 10^3\right)\left(20 \times 10^{-6}\right)=2\ s(b) Write the formulas for charging voltage, substitute values, and solve for v_{C} and v_{R}.
v_C=V\left(1-e^{-t / R C}\right) (19-17)
When t = T = 2 s:
v_C=10\left(1-e^{-2 / 2}\right)=10\left(1-e^{-1}\right)=10(1-0.368)=10(0.632)=6.32\ VV=v_R+v_C (19-15)
v_R=V-v_C=10-6.32=3.68\ VWhen t = 5 s:
v_C=10\left(1-e^{-5 / 2}\right)=10\left(1-e^{-2.5}\right)=10(1-0.082)=10(0.918)=9.18\ Vv_R=V-v_C=10-9.18=0.82\ V
(c) Write the formula for discharging voltage, substitute values, and solve for v_{C} and v_{R}.
v_C=V e^{-t / R C} (19-22)
When t = T = 2 s:
v_C=V e^{-2 / 2}=V e^{-1}=10(0.368)=3.68\ V0=v_R+v_C (19-20)
v_R=-v_C=-3.68\ V