RC Circuits — Question 4

Question 4

The capacitor sees a parallel resistance when a divider source is deactivated

A 12.0 V source feeds R1=3.00 kilohm to node X. R2=6.00 kilohm and C=50.0 microfarad each connect X to ground. The capacitor is initially uncharged.

1. Find the final capacitor voltage and time constant, then vC(t).

2. Find source current immediately after connection and long afterward. Explain why R1+R2 is not the resistance controlling the transient.

Question 4 - Solution

The capacitor sees a parallel resistance when a divider source is deactivated

(a) Method and calculation

$$V_\infty=12\frac6{3+6}=8.00\,\mathrm V,\quad R_{\mathrm{th}}=R_1\parallel R_2=2.00\,\mathrm{k\Omega}.$$$$\tau=R_{\mathrm{th}}C=.100\,\mathrm s,\quad v_C(t)=8(1-e^{-t/.100\,\mathrm s})\ \mathrm V.$$

(b) Interpretation and further calculation

$$I_s(0^+)=(12-0)/3000=4.00\,\mathrm{mA}.$$$$I_s(\infty)=(12-8)/3000=4/3\,\mathrm{mA}.$$Shorting the ideal source makes both resistors connect from X to ground, so they are in parallel as seen by C.

Physical check

The initial capacitor behaves as a zero-voltage element; at steady state it draws no current.