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title: "An ideal LC circuit consists of an inductor with self-inductance \\(L\\) connected to a parallel-plate capacitor with plate area \\(A\\). An external mechanism pulls the capacitor plates apart such that the plate separation increases linearly with time according to \\(d(t) = d_0 (1 + \\alpha t)\\), where \\(d_0\\) and \\(\\alpha\\) are positive constants. Let \\(C_0 = \\dfrac{\\varepsilon_0 A}{d_0}\\) represent the capacitance at time \\(t = 0\\). Assuming electrostatic and magnetostatic approximations hold throughout the motion, which of the following differential equations correctly describes the charge \\(q(t)\\) on the capacitor as a function of time \\(t\\)?"
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url: "https://nerd-notes.com/ubq/118740/"
date_modified: "2026-08-04T08:14:03+00:00"
---

# An ideal LC circuit consists of an inductor with self-inductance \(L\) connected to a parallel-plate capacitor with plate area \(A\). An external mechanism pulls the capacitor plates apart such that the plate separation increases linearly with time according to \(d(t) = d_0 (1 + \alpha t)\), where \(d_0\) and \(\alpha\) are positive constants. Let \(C_0 = \dfrac{\varepsilon_0 A}{d_0}\) represent the capacitance at time \(t = 0\). Assuming electrostatic and magnetostatic approximations hold throughout the motion, which of the following differential equations correctly describes the charge \(q(t)\) on the capacitor as a function of time \(t\)?

An ideal LC circuit consists of an inductor with self-inductance \(L\) connected to a parallel-plate capacitor with plate area \(A\). An external mechanism pulls the capacitor plates apart such that the plate separation increases linearly with time according to \(d(t) = d_0 (1 + \alpha t)\), where \(d_0\) and \(\alpha\) are positive constants. Let \(C_0 = \dfrac{\varepsilon_0 A}{d_0}\) represent the capacitance at time \(t = 0\). Assuming electrostatic and magnetostatic approximations hold throughout the motion, which of the following differential equations correctly describes the charge \(q(t)\) on the capacitor as a function of time \(t\)?

![A schematic diagram of an LC circuit. On the left side, an inductor labeled L is connected in a closed loop to a parallel-plate capacitor on the right side. The capacitor consists of two horizontal parallel metal plates. The upper plate has an upward arrow indicating motion away from the lower plate, with plate separation labeled d(t) = d_0(1 + \alpha t). The upper plate carries charge +q and the lower plate carries charge -q. No other labels, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831243-aynZNJ.jpg)

- **A.** \(\dfrac{d^2 q}{dt^2} + \dfrac{1}{L C_0 (1 + \alpha t)} q = 0\)
- **B.** \(\dfrac{d^2 q}{dt^2} + \dfrac{\alpha}{1 + \alpha t} \dfrac{dq}{dt} + \dfrac{1}{L C_0} q = 0\)
- **C.** \(\dfrac{d^2 q}{dt^2} + \dfrac{\alpha}{1 + \alpha t} \dfrac{dq}{dt} + \dfrac{1 + \alpha t}{L C_0} q = 0\)
- **D.** \(\dfrac{d^2 q}{dt^2} + \dfrac{1 + \alpha t}{L C_0} q = 0\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118740/*