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title: "A capacitor of capacitance \\(C\\) is initially charged to a potential difference \\(V_0\\). At time \\(t = 0\\), switch \\(S\\) is closed, allowing the capacitor to discharge through an ideal resistor of resistance \\(R\\). Which of the following expressions correctly represents the total thermal energy dissipated by the resistor from \\(t = 0\\) to \\(t = 2\\tau\\), where \\(\\tau = RC\\) is the time constant of the circuit?"
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url: "https://nerd-notes.com/ubq/121575/"
date_modified: "2026-08-23T05:24:07+00:00"
---

# A capacitor of capacitance \(C\) is initially charged to a potential difference \(V_0\). At time \(t = 0\), switch \(S\) is closed, allowing the capacitor to discharge through an ideal resistor of resistance \(R\). Which of the following expressions correctly represents the total thermal energy dissipated by the resistor from \(t = 0\) to \(t = 2\tau\), where \(\tau = RC\) is the time constant of the circuit?

A capacitor of capacitance \(C\) is initially charged to a potential difference \(V_0\). At time \(t = 0\), switch \(S\) is closed, allowing the capacitor to discharge through an ideal resistor of resistance \(R\). Which of the following expressions correctly represents the total thermal energy dissipated by the resistor from \(t = 0\) to \(t = 2\tau\), where \(\tau = RC\) is the time constant of the circuit?

![A single rectangular circuit loop oriented with horizontal top and bottom wires and vertical left and right branches. The left branch contains a capacitor represented by two parallel horizontal plates labeled C. The top branch contains an open switch labeled S, angled upward to the right. The right branch contains a resistor represented by a zig-zag line labeled R. The bottom branch is a straight unbroken wire connecting the capacitor to the resistor. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-circuit-1-1787462647-pNie6m.jpg)

- **A.** \(\displaystyle \int_0^{2RC} \dfrac{V_0^2}{R} e^{-t/(RC)} \, dt\)
- **B.** \(\displaystyle \int_0^{2RC} \dfrac{V_0^2}{R} \left(1 - e^{-t/(RC)}\right)^2 dt\)
- **C.** \(\displaystyle \int_0^{2RC} \dfrac{V_0^2}{R} e^{-2t/(RC)} \, dt\)
- **D.** \(\displaystyle \int_0^{2RC} \dfrac{V_0^2}{2R} e^{-2t/(RC)} \, dt\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121575/*