---
title: "A circuit contains an ideal battery of electromotive force \\(\\mathcal{E}\\), an open switch, a resistor of resistance \\(R\\), and an initially uncharged capacitor of capacitance \\(C\\) connected in series. At time \\(t = 0\\), the switch is closed and the capacitor is allowed to charge completely until current ceases. If the experiment is repeated using an identical battery and capacitor but with a resistor of resistance \\(2R\\), how does the total thermal energy dissipated in the resistor compare to that in the original circuit, and why?"
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url: "https://nerd-notes.com/ubq/121580/"
date_modified: "2026-08-23T05:24:09+00:00"
---

# A circuit contains an ideal battery of electromotive force \(\mathcal{E}\), an open switch, a resistor of resistance \(R\), and an initially uncharged capacitor of capacitance \(C\) connected in series. At time \(t = 0\), the switch is closed and the capacitor is allowed to charge completely until current ceases. If the experiment is repeated using an identical battery and capacitor but with a resistor of resistance \(2R\), how does the total thermal energy dissipated in the resistor compare to that in the original circuit, and why?

A circuit contains an ideal battery of electromotive force \(\mathcal{E}\), an open switch, a resistor of resistance \(R\), and an initially uncharged capacitor of capacitance \(C\) connected in series. At time \(t = 0\), the switch is closed and the capacitor is allowed to charge completely until current ceases. If the experiment is repeated using an identical battery and capacitor but with a resistor of resistance \(2R\), how does the total thermal energy dissipated in the resistor compare to that in the original circuit, and why?

![A single-loop rectangular circuit diagram drawn with thin solid black lines. On the left vertical segment is an ideal battery labeled \(\mathcal{E}\), with a longer horizontal line above a shorter parallel horizontal line. On the top horizontal segment is an open single-pole single-throw switch labeled \(S\) followed to the right by a zigzag resistor labeled \(R\). On the right vertical segment is a capacitor labeled \(C\), represented by two parallel horizontal lines of equal length. The bottom horizontal segment consists of a continuous wire returning to the battery. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787462649-CraC9B.jpg)

- **A.** The total thermal energy dissipated is the same, because the total work done by the battery is \(W = C\mathcal{E}^2\) and the final energy stored in the capacitor is \(U_C = \dfrac{1}{2}C\mathcal{E}^2\), requiring \(\dfrac{1}{2}C\mathcal{E}^2\) to be dissipated regardless of \(R\).
- **B.** The total thermal energy dissipated is greater, because doubling the resistance doubles the time constant to \(\tau = 2RC\), causing current to flow through a larger resistance for a longer duration.
- **C.** The total thermal energy dissipated is less, because the maximum rate of Joule heating is reduced from \(\dfrac{\mathcal{E}^2}{R}\) to \(\dfrac{\mathcal{E}^2}{2R}\), reducing the total integrated power over time.
- **D.** The total thermal energy dissipated is less, because a larger resistance prevents the capacitor from reaching full charge, thereby decreasing the total energy supplied by the battery.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121580/*