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title: "A capacitor of capacitance \\(C\\) has an initial charge \\(Q_0\\) on its top plate and \\(-Q_0\\) on its bottom plate. It is connected in a circuit with an open switch \\(S\\) and an ideal inductor of inductance \\(L\\), as shown in Figure 1. The resistance of the wires is negligible. At time \\(t = 0\\), switch \\(S\\) is closed. Let the clockwise direction be the positive direction for current \\(I\\), such that the current is related to the charge \\(q\\) on the top plate by \\(I = -\\dfrac{dq}{dt}\\)."
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url: "https://nerd-notes.com/ubq/117867/"
date_modified: "2026-08-04T08:00:07+00:00"
---

# A capacitor of capacitance \(C\) has an initial charge \(Q_0\) on its top plate and \(-Q_0\) on its bottom plate. It is connected in a circuit with an open switch \(S\) and an ideal inductor of inductance \(L\), as shown in Figure 1. The resistance of the wires is negligible. At time \(t = 0\), switch \(S\) is closed. Let the clockwise direction be the positive direction for current \(I\), such that the current is related to the charge \(q\) on the top plate by \(I = -\dfrac{dq}{dt}\).

A capacitor of capacitance \(C\) has an initial charge \(Q_0\) on its top plate and \(-Q_0\) on its bottom plate. It is connected in a circuit with an open switch \(S\) and an ideal inductor of inductance \(L\), as shown in Figure 1. The resistance of the wires is negligible. At time \(t = 0\), switch \(S\) is closed. Let the clockwise direction be the positive direction for current \(I\), such that the current is related to the charge \(q\) on the top plate by \(I = -\dfrac{dq}{dt}\).

![A rectangular circuit diagram. On the left vertical wire, there is a capacitor symbol labeled \(C\). The top plate of the capacitor is labeled with a plus sign and the bottom plate with a minus sign. On the right vertical wire, there is an inductor symbol, drawn as a coil, labeled \(L\). On the top horizontal wire, there is an open switch labeled \(S\). Inside the loop, there is a curved arrow pointing clockwise labeled \(I\). No other components, lines, or text appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830406-koJC6g.jpg)

**Part a)** Using Kirchhoff's loop rule, **derive** a differential equation for the charge \(q\) on the top plate of the capacitor as a function of time \(t\). Express your answer in terms of \(q\), \(C\), \(L\), and physical constants, as appropriate.

**Part b)** On the axes below, **sketch** graphs of the charge \(q\) on the top plate of the capacitor and the current \(I\) in the circuit as functions of time \(t\) for two full cycles of oscillation. Explicitly **label** any maximum or minimum values with algebraic expressions in terms of \(Q_0\), \(C\), and \(L\).

**Part c)** On the axes below, **sketch** graphs of the electrical energy \(U_C\) stored in the capacitor and the magnetic energy \(U_L\) stored in the inductor as functions of time \(t\) for the same two full cycles of oscillation. Clearly **label** each curve.

**Part d)** The physical behavior of an ideal LC circuit is mathematically analogous to the simple harmonic motion of a mechanical system, such as a block of mass \(m\) oscillating on a horizontal spring of force constant \(k\).

**Part e)** Suppose the ideal wires in the circuit are replaced with wires that have a small, non-negligible constant total resistance \(R\). The capacitor is again charged to \(Q_0\) and the switch is closed at \(t = 0\).


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117867/*