---
title: "A circuit is constructed with an ideal battery of electromotive force (emf) \\(\\varepsilon\\), a switch \\(S\\), an initially uncharged capacitor of capacitance \\(C\\), and three resistors with resistances \\(R_1 = R\\), \\(R_2 = 2R\\), and \\(R_3 = R\\). The components are connected as shown in Figure 1."
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/117743/"
date_modified: "2026-08-04T07:55:19+00:00"
---

# A circuit is constructed with an ideal battery of electromotive force (emf) \(\varepsilon\), a switch \(S\), an initially uncharged capacitor of capacitance \(C\), and three resistors with resistances \(R_1 = R\), \(R_2 = 2R\), and \(R_3 = R\). The components are connected as shown in Figure 1.

A circuit is constructed with an ideal battery of electromotive force (emf) \(\varepsilon\), a switch \(S\), an initially uncharged capacitor of capacitance \(C\), and three resistors with resistances \(R_1 = R\), \(R_2 = 2R\), and \(R_3 = R\). The components are connected as shown in Figure 1.

![A rectangular circuit diagram. On the left branch is a battery with emf \(\varepsilon\), with the longer positive terminal pointing up. The top branch contains a switch \(S\) and a resistor \(R_1\) to its right. To the right of \(R_1\), the circuit splits into two parallel vertical branches. The middle vertical branch contains resistor \(R_2\). The rightmost vertical branch contains a capacitor \(C\) and a resistor \(R_3\) in series. The bottom branch connects the bottoms of the middle and right vertical branches back to the negative terminal of the battery. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830118-Ld7ECW.jpg)

**Part a)** At time \(t = 0\), switch \(S\) is closed.

**Part b)** On the axes provided, **sketch** a graph of the current \(I_1\) through resistor \(R_1\) and the current \(I_3\) through resistor \(R_3\) as functions of time \(t\) from \(t = 0\) until a long time later. - Label the vertical axis with appropriate values in terms of \(\varepsilon\) and \(R\) to indicate the initial and steady-state currents. - Clearly distinguish the two curves by labeling them "\(I_1\)" and "\(I_3\)".

**Part c)** After a long time, the capacitor is fully charged. At a new time \(t = t_1\), switch \(S\) is opened.

**Part d)** **Derive** an expression for the potential difference \(\Delta V_C(t')\) across the capacitor as a function of the time \(t'\) elapsed since switch \(S\) was opened. Express your answer in terms of \(\varepsilon\), \(R\), \(C\), \(t'\), and fundamental constants.


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117743/*