RC Circuits — Question 1

Question 1

A precharged capacitor approaches a new source voltage from its actual initial value

At t=0, a capacitor C=100 microfarad initially at vC=3.00 V is connected through R=2.00 kilohm to an ideal 12.0 V source. Voltage vC is positive on the upper capacitor plate; current i enters that plate.

1. Derive vC(t) and i(t), including their initial values.

2. Find the first time vC reaches 9.00 V and the current then.

Question 1 - Solution

A precharged capacitor approaches a new source voltage from its actual initial value

(a) Method and calculation

$$RC\frac{dv_C}{dt}+v_C=E,\quad\tau=RC=.200\,\mathrm s.$$$$v_C=12-9e^{-t/\tau}\ \mathrm V,\quad i=C\dot v_C=4.50e^{-t/\tau}\ \mathrm{mA}.$$

(b) Interpretation and further calculation

$$v_C(0)=3.00\,\mathrm V,\quad i(0)=4.50\,\mathrm{mA},\quad e^{-t/\tau}=(12-9)/9=1/3.$$$$t=\tau\ln3=.21972\,\mathrm s,\quad i=(12-9)/2000=1.50\,\mathrm{mA}.$$

Physical check

Capacitor voltage is continuous through switching with finite resistance; its derivative and current can change abruptly.