---
title: "A circuit is constructed with an ideal battery of emf \\(\\varepsilon\\), two resistors with resistances \\(R_1\\) and \\(R_2\\), an ideal inductor of inductance \\(L\\), and a switch \\(S\\), as shown in Figure 1. The switch \\(S\\) is initially open, and there is no current in the inductor."
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/117878/"
date_modified: "2026-08-04T08:00:16+00:00"
---

# A circuit is constructed with an ideal battery of emf \(\varepsilon\), two resistors with resistances \(R_1\) and \(R_2\), an ideal inductor of inductance \(L\), and a switch \(S\), as shown in Figure 1. The switch \(S\) is initially open, and there is no current in the inductor.

A circuit is constructed with an ideal battery of emf \(\varepsilon\), two resistors with resistances \(R_1\) and \(R_2\), an ideal inductor of inductance \(L\), and a switch \(S\), as shown in Figure 1. The switch \(S\) is initially open, and there is no current in the inductor.

![A rectangular circuit diagram. The left vertical wire contains a battery with the long line (positive terminal) on top, labeled '\(\varepsilon\)'. The top horizontal wire goes right from the battery, contains an open switch labeled '\(S\)', and then a resistor labeled '\(R_1\)'. After \(R_1\), the top wire meets a junction. A middle vertical wire connects this top junction to a bottom junction and contains a resistor labeled '\(R_2\)'. The top wire continues right from the top junction, turns down into a right vertical wire containing an inductor labeled '\(L\)', and then turns left to meet the bottom junction. The bottom horizontal wire connects the bottom junction, the middle junction, and the battery's negative terminal. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830415-aAso2q.jpg)

**Part a)** The switch \(S\) is closed at time \(t=0\). Express your answers in terms of \(\varepsilon\), \(R_1\), \(R_2\), \(L\), and physical constants, as appropriate. *(2 points)*

**Part b)** **Sketch** a graph of the magnitude of the potential difference across the inductor, \(\Delta V_L\), as a function of time \(t\) from \(t=0\) until a long time later. The value \(\Delta V_{L0}\) represents the potential difference immediately after the switch is closed. *(2 points)*

**Part c)** A student claims that the potential difference across the inductor decreases over time because the rate of change of the current through the inductor is decreasing. **Justify** the student's claim using the physical principles of inductance, and **explain** how the student's claim is consistent with your graph in part (b). *(2 points)*

**Part d)** After the switch has been closed for a long time, it is suddenly opened. Let this be a new time \(t=0\). *(5 points)*

**Part e)** A student observes that if \(R_2\) is much greater than \(R_1\), a spark can jump across the contacts of switch \(S\) at the instant it is opened. **Explain** how the expression derived in part (d)(iii) supports this observation by considering the potential difference across the switch. *(2 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117878/*