Chapter 26 Problem 49

Problem

The RC circuit

In this circuit all resistors have the same resistance R. At $t=0$ with the capacitor C uncharged, the switch is closed.

(a) At $t=0$ the three currents can be determined by analyzing a simpler, but equivalent, circuit. Identify this simpler circuit and use it to find the values of $I_{1}$, $I_{2}$ and $I_{3}$ at $t=0$

(b) At the currents can be determined by analyzing a simpler, equivalent circuit. Identify this simpler circuit and implement it in finding the values of $I_{1}$, $I_{2}$ and $I_{3}$ at $t=\infty$

(c) At $t=\infty$ what is the potential difference across the capacitor?

Solution

(a)

At $t=0$ capacitor is "hungry" for charge, and can be considered as a short. Thus the simpler circuit is simply a circuit with capacitor removed.

$R_{\textrm {eq}}=R+\left({\frac {1}{R}}+{\frac {1}{R}}\right)^{-1}={\frac {3}{2}}R$ $\rightarrow I_{1}={\frac {\mathcal {E}}{R_{\textrm {eq}}}}={\frac {2{\mathcal {E}}}{3R}}$

$\rightarrow I_{2}=I_{3}={\frac {1}{2}}I_{1}={\frac {\mathcal {E}}{3R}}$

(b)

At $t=\infty$ capacitor is "full" for charge, and no current will flow through it. Thus the simpler circuit is simply a circuit with the branch with capacitor removed.

$R_{\textrm {eq}}=R+R=2R$

$\rightarrow I_{1}=I_{2}={\frac {\mathcal {E}}{R_{\textrm {eq}}}}={\frac {\mathcal {E}}{2R}}$

$I_{3}=0$

(c)

At $t=\infty$ capacitor is "full" for charge, and no current will flow through it. Thus the voltage drop through $I_{3}$ branch is 0. Thus the voltage drop in the capacitor is equal to voltage drop in $I_{2}$ branch.

$V_{C}=V_{R_{2}}=I_{2}R={\frac {\mathcal {E}}{2R}}R={\frac {1}{2}}{\mathcal {E}}$