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title: "A circuit contains an ideal battery of emf \\(\\mathcal{E}\\), two open switches \\(S_1\\) and \\(S_2\\), two resistors of resistances \\(R_1\\) and \\(R_2\\), and an initially uncharged capacitor of capacitance \\(C\\). Switch \\(S_1\\) and resistor \\(R_1\\) are in series with the battery in the main loop, while the capacitor \\(C\\) is connected in parallel with a branch containing switch \\(S_2\\) and resistor \\(R_2\\). At time \\(t = 0\\), switches \\(S_1\\) and \\(S_2\\) are closed simultaneously. Which of the following correctly identifies the initial current \\(I_0\\) supplied by the battery immediately after the switches are closed, the time constant \\(\\tau\\) of the charging circuit, and the steady-state charge \\(Q_{\\infty}\\) stored on the capacitor as \\(t \\to \\infty\\)?"
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url: "https://nerd-notes.com/ubq/124963/"
date_modified: "2026-09-28T14:12:16+00:00"
---

# A circuit contains an ideal battery of emf \(\mathcal{E}\), two open switches \(S_1\) and \(S_2\), two resistors of resistances \(R_1\) and \(R_2\), and an initially uncharged capacitor of capacitance \(C\). Switch \(S_1\) and resistor \(R_1\) are in series with the battery in the main loop, while the capacitor \(C\) is connected in parallel with a branch containing switch \(S_2\) and resistor \(R_2\). At time \(t = 0\), switches \(S_1\) and \(S_2\) are closed simultaneously. Which of the following correctly identifies the initial current \(I_0\) supplied by the battery immediately after the switches are closed, the time constant \(\tau\) of the charging circuit, and the steady-state charge \(Q_{\infty}\) stored on the capacitor as \(t \to \infty\)?

A circuit contains an ideal battery of emf \(\mathcal{E}\), two open switches \(S_1\) and \(S_2\), two resistors of resistances \(R_1\) and \(R_2\), and an initially uncharged capacitor of capacitance \(C\). Switch \(S_1\) and resistor \(R_1\) are in series with the battery in the main loop, while the capacitor \(C\) is connected in parallel with a branch containing switch \(S_2\) and resistor \(R_2\). At time \(t = 0\), switches \(S_1\) and \(S_2\) are closed simultaneously. Which of the following correctly identifies the initial current \(I_0\) supplied by the battery immediately after the switches are closed, the time constant \(\tau\) of the charging circuit, and the steady-state charge \(Q_{\infty}\) stored on the capacitor as \(t \to \infty\)?

![A rectangular circuit schematic oriented horizontally. On the left vertical wire is a DC voltage source labeled \(\mathcal{E}\), with the longer horizontal line on top and shorter thicker line on bottom. The top horizontal wire runs to the right, containing an open switch labeled \(S_1\) followed by a resistor labeled \(R_1\). To the right of resistor \(R_1\), the wire splits at a junction into two parallel vertical branches that connect to a bottom horizontal return wire. The first vertical branch contains an uncharged parallel-plate capacitor labeled \(C\). The second vertical branch, positioned to the right of the first, contains an open switch labeled \(S_2\) in series with a resistor labeled \(R_2\). The bottom horizontal wire connects the bottoms of both vertical branches straight back to the negative terminal of the voltage source. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604736-t1zPBF.jpg)

- **A.** \(I_0 = \dfrac{\mathcal{E}}{R_1 + R_2}\) ; \quad \tau = (R_1 + R_2)C ; \quad Q_{\infty} = C\mathcal{E}
- **B.** \(I_0 = \dfrac{\mathcal{E}}{R_1}\) ; \quad \tau = \left(\dfrac{R_1 R_2}{R_1 + R_2}\right)C ; \quad Q_{\infty} = C\mathcal{E}\left(\dfrac{R_2}{R_1 + R_2}\right)
- **C.** \(I_0 = \dfrac{\mathcal{E}}{R_1}\) ; \quad \tau = R_1 C ; \quad Q_{\infty} = C\mathcal{E}\left(\dfrac{R_1}{R_1 + R_2}\right)
- **D.** \(I_0 = \dfrac{\mathcal{E}}{R_1}\) ; \quad \tau = (R_1 + R_2)C ; \quad Q_{\infty} = C\mathcal{E}\left(\dfrac{R_2}{R_1 + R_2}\right)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124963/*