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title: "An uncharged parallel-plate capacitor of capacitance \\(C\\), a resistor of resistance \\(R\\), an ideal battery of potential difference \\(\\Delta V\\), and an open switch are connected in series, as shown in the circuit diagram. At time \\(t = 0\\), the switch is closed. Which of the following correctly compares the energy \\(U_C\\) stored in the capacitor long after the switch is closed (\\(t \\to \\infty\\)) to the total energy \\(E_{\\text{batt}}\\rangle\\) supplied by the battery during this process, and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/117210/"
date_modified: "2026-08-04T06:44:59+00:00"
---

# An uncharged parallel-plate capacitor of capacitance \(C\), a resistor of resistance \(R\), an ideal battery of potential difference \(\Delta V\), and an open switch are connected in series, as shown in the circuit diagram. At time \(t = 0\), the switch is closed. Which of the following correctly compares the energy \(U_C\) stored in the capacitor long after the switch is closed (\(t \to \infty\)) to the total energy \(E_{\text{batt}}\rangle\) supplied by the battery during this process, and provides the correct physical justification?

An uncharged parallel-plate capacitor of capacitance \(C\), a resistor of resistance \(R\), an ideal battery of potential difference \(\Delta V\), and an open switch are connected in series, as shown in the circuit diagram. At time \(t = 0\), the switch is closed. Which of the following correctly compares the energy \(U_C\) stored in the capacitor long after the switch is closed (\(t \to \infty\)) to the total energy \(E_{\text{batt}}\rangle\) supplied by the battery during this process, and provides the correct physical justification?

![A rectangular schematic diagram of a single-loop series circuit. The bottom horizontal branch contains an ideal DC battery labeled \(\Delta V\) with its longer positive plate on the left and shorter negative plate on the right. The left vertical branch contains an open switch labeled \(S\). The top horizontal branch contains a resistor labeled \(R\) represented by a zigzag line. The right vertical branch contains a parallel-plate capacitor labeled \(C\) represented by two equal parallel vertical lines. Standard thin lines connect all components in a single closed loop. No other labels, lines, or text appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785825899-hw91Md.jpg)

- **A.** The energy stored in the capacitor is equal to the total energy supplied by the battery (\(U_C = E_{\text{batt}}\)) because energy is conserved and no charge is lost from the circuit as the capacitor reaches full charge.
- **B.** The energy stored in the capacitor is equal to the total energy supplied by the battery (\(U_C = E_{\text{batt}}\)) because the potential difference across the capacitor eventually becomes equal to the battery potential difference \(\Delta V\).
- **C.** The energy stored in the capacitor is half the total energy supplied by the battery (\(U_C = \frac{1}{2} E_{\text{batt}}\)) because the current through the resistor decreases linearly to zero halfway through the charging process.
- **D.** The energy stored in the capacitor is half the total energy supplied by the battery (\(U_C = \frac{1}{2} E_{\text{batt}}\)) because half of the energy delivered by the battery is dissipated as thermal energy in the resistor while charge flows.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117210/*