---
title: "An ideal battery of electromotive force \\(\\mathcal{E}\\) is connected in series at time \\(t = 0\\) with an ideal switch S, a resistor of resistance \\(R\\), and an inductor of inductance \\(L\\). At a specific time \\(t_1\\), the rate at which thermal energy is dissipated in the resistor equals the rate at which energy is being stored in the magnetic field of the inductor. What are the potential difference across the inductor \\(V_L\\) and the time \\(t_1\\) at which this equality occurs?"
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/118719/"
date_modified: "2026-08-04T08:13:52+00:00"
---

# An ideal battery of electromotive force \(\mathcal{E}\) is connected in series at time \(t = 0\) with an ideal switch S, a resistor of resistance \(R\), and an inductor of inductance \(L\). At a specific time \(t_1\), the rate at which thermal energy is dissipated in the resistor equals the rate at which energy is being stored in the magnetic field of the inductor. What are the potential difference across the inductor \(V_L\) and the time \(t_1\) at which this equality occurs?

An ideal battery of electromotive force \(\mathcal{E}\) is connected in series at time \(t = 0\) with an ideal switch S, a resistor of resistance \(R\), and an inductor of inductance \(L\). At a specific time \(t_1\), the rate at which thermal energy is dissipated in the resistor equals the rate at which energy is being stored in the magnetic field of the inductor. What are the potential difference across the inductor \(V_L\) and the time \(t_1\) at which this equality occurs?

![A single-loop rectangular schematic diagram of an LR circuit oriented horizontally. The left vertical branch contains a DC voltage source labeled \(\mathcal{E}\) with the longer positive terminal line at the top and shorter negative terminal line at the bottom. The top horizontal wire contains an open single-pole single-throw switch labeled S. The right vertical branch contains a resistor drawn as a standard zig-zag line labeled R. The bottom horizontal wire contains an inductor drawn as a series of four curved semi-circular loops labeled L. Straight solid lines connect all circuit elements into a single closed rectangular loop. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831232-mht92x.jpg)

- **A.** \(V_L = \dfrac{\mathcal{E}}{e}\) and \(t_1 = \dfrac{L}{R}\)
- **B.** \(V_L = \dfrac{\mathcal{E}}{e}\) and \(t_1 = \dfrac{L}{R} \ln 2\)
- **C.** \(V_L = \dfrac{\mathcal{E}}{2}\) and \(t_1 = \dfrac{L}{R}\)
- **D.** \(V_L = \dfrac{\mathcal{E}}{2}\) and \(t_1 = \dfrac{L}{R} \ln 2\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118719/*