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title: "A real capacitor is modeled as an ideal capacitor of capacitance \\(C\\) connected in parallel with an internal leakage resistance \\(R_L\\).  This parallel combination is placed in series with an ideal battery of electromotive force \\(\\mathcal{E}\\), an open switch \\(S\\), and an external resistor of resistance \\(R\\).  At time \\(t = 0\\), the switch is closed with the capacitor initially uncharged, resulting in a time-dependent charge of \\(q(t) = \\dfrac{C\\mathcal{E} R_L}{R + R_L}\\left(1 – e^{-t/\\tau}\\right)\\), where \\(\\tau = \\left(\\dfrac{R R_L}{R + R_L}\\right)C\\).  Which of the following statements correctly describes the physical behavior of the circuit in the specified limit?"
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url: "https://nerd-notes.com/ubq/124880/"
date_modified: "2026-09-28T14:11:47+00:00"
---

# A real capacitor is modeled as an ideal capacitor of capacitance \(C\) connected in parallel with an internal leakage resistance \(R_L\).

This parallel combination is placed in series with an ideal battery of electromotive force \(\mathcal{E}\), an open switch \(S\), and an external resistor of resistance \(R\).

At time \(t = 0\), the switch is closed with the capacitor initially uncharged, resulting in a time-dependent charge of \(q(t) = \dfrac{C\mathcal{E} R_L}{R + R_L}\left(1 – e^{-t/\tau}\right)\), where \(\tau = \left(\dfrac{R R_L}{R + R_L}\right)C\).

Which of the following statements correctly describes the physical behavior of the circuit in the specified limit?

A real capacitor is modeled as an ideal capacitor of capacitance \(C\) connected in parallel with an internal leakage resistance \(R_L\).

This parallel combination is placed in series with an ideal battery of electromotive force \(\mathcal{E}\), an open switch \(S\), and an external resistor of resistance \(R\).

At time \(t = 0\), the switch is closed with the capacitor initially uncharged, resulting in a time-dependent charge of \(q(t) = \dfrac{C\mathcal{E} R_L}{R + R_L}\left(1 - e^{-t/\tau}\right)\), where \(\tau = \left(\dfrac{R R_L}{R + R_L}\right)C\).

Which of the following statements correctly describes the physical behavior of the circuit in the specified limit?

![A single rectangular circuit schematic oriented horizontally. On the left vertical wire, a battery symbol is shown with a long line on top and a short thick line on bottom, labeled \(\mathcal{E}\) to its left. The wire continues upward to a top horizontal branch containing an open knife switch labeled \(S\) and a series resistor drawn as a zigzag segment labeled \(R\). The wire then splits at a right-hand junction into two parallel vertical branches before recombining at a bottom junction. The left parallel branch contains an ideal capacitor drawn as two parallel horizontal plates labeled \(C\). The right parallel branch contains a resistor drawn as a zigzag segment labeled \(R_L\). The recombined bottom wire runs horizontally back to the bottom of the battery. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-circuit-1-1790604707-ovXdF0.jpg)

- **A.** In the limit \(R_L \to \infty\), the effective time constant approaches \(\tau \to \infty\), preventing the capacitor from ever reaching a steady-state charge.
- **B.** In the limit \(t \to \infty\), the rate of total energy dissipation in the circuit approaches zero because the fully charged capacitor branch blocks all current from the battery.
- **C.** In the limit \(t \to \infty\), the steady-state charge on the capacitor is strictly less than \(C\mathcal{E}\) because a continuous current flows through \(R_L\), maintaining a non-zero potential drop across \(R\).
- **D.** In the limit \(R_L \to 0\), the time constant approaches \(\tau \to 0\), causing the capacitor to charge to a maximum value of \(C\mathcal{E}\) instantaneously.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124880/*