RC Circuits — Question 5

Question 5

A two-stage switch preserves capacitor voltage but changes the exponential law

An initially uncharged capacitor charges from a 10.0 V source through R1 for $T=R_1C\ln2$. The source is then disconnected and the capacitor discharges through R2=2R1 only. Let tau=R1C and i=C dvC/dt.

1. Write vC(t) before and after T, enforcing continuity.

2. Find the additional time after T to reach 1.00 V and compare i immediately before and after switching.

Question 5 - Solution

A two-stage switch preserves capacitor voltage but changes the exponential law

(a) Method and calculation

$$0\le t<T:\quad v_C=10(1-e^{-t/\tau})\ \mathrm V,\quad v_C(T)=5.00\,\mathrm V.$$$$t\ge T:\quad v_C=5e^{-(t-T)/(2\tau)}\ \mathrm V.$$

(b) Interpretation and further calculation

$$1=5e^{-\Delta t/(2\tau)}\Rightarrow\Delta t=2\tau\ln5.$$$$i(T^-)=(5.00\,\mathrm V)/R_1,\quad i(T^+)=-(5.00\,\mathrm V)/(2R_1).$$

Physical check

An ideal switch can change current discontinuously while the capacitor voltage remains continuous.