(II) Consider the circuit shown in Fig. 26–67, where all resistors have the same resistance R. At t = 0, with the capacitor C uncharged, the switch is closed. At t = ∞, the currents can be determined by analyzing a simpler, equivalent circuit. Identify this simpler circuit and implement it in finding the values of I₁, I₂ and I₃ at t = ∞.

Verified step by step guidance

1. Step 1: At t = ∞, the capacitor C becomes fully charged, and no current flows through it. This means the branch containing the capacitor can be ignored when analyzing the circuit. The circuit simplifies to a series-parallel combination of resistors.
2. Step 2: Identify the equivalent circuit. The resistor R in series with the battery remains unchanged. The two resistors in parallel (one in the middle branch and one in the branch with the capacitor) form an equivalent resistance. Use the formula for parallel resistance: $\frac{1}{\mathrm{R}+\mathrm{R}}$.
3. Step 3: Calculate the total resistance of the circuit. Add the equivalent resistance of the parallel combination to the resistance of the resistor in series with the battery: $\mathrm{R}+\frac{\mathrm{R}}{2}$.
4. Step 4: Use Ohm's Law to find the total current I supplied by the battery: $\frac{\epsilon }{\mathrm{R}+\frac{\mathrm{R}}{2}}$, where ε is the emf of the battery.
5. Step 5: Determine the currents I1, I2, and I3. I1 is the current through the resistor in series with the battery, which equals the total current I. I2 and I3 are the currents through the parallel resistors, which can be calculated using the current division rule: $\frac{\mathrm{R}}{\mathrm{R}+\mathrm{R}}$ for each branch.

Key Concepts

Kirchhoff's Laws

Kirchhoff's Laws consist of two fundamental principles for analyzing electrical circuits: Kirchhoff's Current Law (KCL) states that the total current entering a junction equals the total current leaving it, while Kirchhoff's Voltage Law (KVL) asserts that the sum of the electrical potential differences (voltage) around any closed circuit loop must equal zero. These laws are essential for determining current and voltage in complex circuits.

Capacitor Charging and Steady State

When a capacitor is connected in a circuit, it initially allows current to flow as it charges. Over time, as it reaches full charge, the current through the capacitor decreases to zero, and it behaves like an open circuit. At steady state (t = ∞), the capacitor is fully charged, and the analysis simplifies to the resistive components of the circuit, allowing for easier calculation of currents.

Equivalent Resistance

Equivalent resistance is a simplified representation of a complex circuit that allows for easier calculations of current and voltage. For resistors in series, the equivalent resistance is the sum of individual resistances, while for resistors in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of the individual resistances. This concept is crucial for analyzing circuits with multiple resistors and determining the total current flowing through the circuit.