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title: "A circuit consists of an ideal battery of constant electromotive force \\(\\mathcal{E}\\), a resistor of resistance \\(R\\), an open switch, and an uncharged non-linear capacitor connected in series. The capacitance of the capacitor depends on the potential difference \\(V\\) across its plates according to \\(C(V) = C_0\\left(1 + \\dfrac{V}{\\mathcal{E}}\\right)\\), where \\(C_0\\) is a positive constant. The switch is closed at time \\(t = 0\\). Which of the following expressions correctly gives the rate of change of the potential difference across the capacitor, \\(\\dfrac{dV}{dt}\\), as a function of \\(V\\)?"
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url: "https://nerd-notes.com/ubq/118427/"
date_modified: "2026-08-04T08:10:01+00:00"
---

# A circuit consists of an ideal battery of constant electromotive force \(\mathcal{E}\), a resistor of resistance \(R\), an open switch, and an uncharged non-linear capacitor connected in series. The capacitance of the capacitor depends on the potential difference \(V\) across its plates according to \(C(V) = C_0\left(1 + \dfrac{V}{\mathcal{E}}\right)\), where \(C_0\) is a positive constant. The switch is closed at time \(t = 0\). Which of the following expressions correctly gives the rate of change of the potential difference across the capacitor, \(\dfrac{dV}{dt}\), as a function of \(V\)?

A circuit consists of an ideal battery of constant electromotive force \(\mathcal{E}\), a resistor of resistance \(R\), an open switch, and an uncharged non-linear capacitor connected in series. The capacitance of the capacitor depends on the potential difference \(V\) across its plates according to \(C(V) = C_0\left(1 + \dfrac{V}{\mathcal{E}}\right)\), where \(C_0\) is a positive constant. The switch is closed at time \(t = 0\). Which of the following expressions correctly gives the rate of change of the potential difference across the capacitor, \(\dfrac{dV}{dt}\), as a function of \(V\)?

![A schematic diagram of a single-loop electric circuit containing a battery of electromotive force \varepsilon on the left vertical branch, a switch at the top horizontal wire, a resistor of resistance R on the right vertical branch, and a non-linear capacitor labeled C(V) on the bottom horizontal branch. The loop is drawn as a rectangle with thin solid lines. The battery is depicted with standard long and short parallel plate symbols, with the longer plate on top labeled with a plus sign. The resistor is depicted with a standard zigzag line labeled R. The capacitor is depicted with two parallel lines labeled C(V) = C_0(1 + V/\varepsilon). No arrows or charge signs appear on the schematic. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831001-7zQ9P9.jpg)

- **A.** \(\dfrac{dV}{dt} = \dfrac{\mathcal{E} - V}{R C_0}\)
- **B.** \(\dfrac{dV}{dt} = \dfrac{\mathcal{E}(\mathcal{E} - V)}{R C_0 (\mathcal{E} + V)}\)
- **C.** \(\dfrac{dV}{dt} = \dfrac{\mathcal{E}(\mathcal{E} + V)}{R C_0 (\mathcal{E} + 2V)}\)
- **D.** \(\dfrac{dV}{dt} = \dfrac{\mathcal{E}(\mathcal{E} - V)}{R C_0 (\mathcal{E} + 2V)}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118427/*