(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 = ∞, what is the potential difference across the capacitor?

Verified step by step guidance

1. Step 1: Analyze the circuit configuration. The circuit consists of a battery with emf \( \mathcal{E} \), three resistors each with resistance \( R \), and a capacitor \( C \). At \( t = 0 \), the capacitor is uncharged, and the switch is closed. At \( t = \infty \), the capacitor becomes fully charged, and the current through the capacitor branch stops flowing.
2. Step 2: Understand the behavior of the capacitor at \( t = \infty \). When the capacitor is fully charged, it acts as an open circuit because no current flows through it. The potential difference across the capacitor will equal the voltage drop across the resistor in the same branch.
3. Step 3: Simplify the circuit at \( t = \infty \). With the capacitor acting as an open circuit, the current flows only through the two resistors in series connected to the battery. The equivalent resistance of these two resistors is \( R_{eq} = R + R = 2R \).
4. Step 4: Calculate the current through the circuit. Using Ohm's law, the total current \( I \) supplied by the battery is \( I = \frac{\mathcal{E}}{R_{eq}} = \frac{\mathcal{E}}{2R} \). This current flows through the first resistor \( R \) and splits at the junction.
5. Step 5: Determine the potential difference across the capacitor. At \( t = \infty \), the voltage across the capacitor is equal to the voltage drop across the resistor in its branch. Using the current \( I \) and Ohm's law, the potential difference across the capacitor is \( V_C = \frac{\mathcal{E}}{2} \).

Key Concepts

Capacitance

Capacitance is the ability of a capacitor to store charge per unit voltage. It is defined as C = Q/V, where C is capacitance, Q is the charge stored, and V is the voltage across the capacitor. In a circuit, when a capacitor is connected, it will charge up to a certain voltage depending on the circuit configuration and the time constant, which is influenced by the resistance and capacitance values.

Steady State in RC Circuits

In an RC (resistor-capacitor) circuit, steady state refers to the condition when the capacitor is fully charged and no current flows through it. This occurs after a long time has passed since the switch was closed, typically considered as t approaching infinity. At this point, the voltage across the capacitor equals the supply voltage, and the current in the circuit drops to zero.

Voltage Division in Series Circuits

In a series circuit, the total voltage is divided among the components based on their resistances. The voltage across each resistor can be calculated using the formula Vx = (Rx / Rtotal) * Vtotal, where Vx is the voltage across the resistor, Rx is its resistance, and Rtotal is the total resistance in the circuit. This principle is crucial for determining the voltage across the capacitor once it reaches steady state.