---
title: "A circuit consists of an ideal battery of constant emf \\(\\mathcal{E}\\), a resistor of resistance \\(R\\), and an inductor whose inductance \\(L(t)\\) varies with time as a ferromagnetic core is pulled out of the coil. Which of the following differential equations correctly describes the current \\(I(t)\\) in the circuit while the core is being pulled?"
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url: "https://nerd-notes.com/ubq/118673/"
date_modified: "2026-08-04T08:13:39+00:00"
---

# A circuit consists of an ideal battery of constant emf \(\mathcal{E}\), a resistor of resistance \(R\), and an inductor whose inductance \(L(t)\) varies with time as a ferromagnetic core is pulled out of the coil. Which of the following differential equations correctly describes the current \(I(t)\) in the circuit while the core is being pulled?

A circuit consists of an ideal battery of constant emf \(\mathcal{E}\), a resistor of resistance \(R\), and an inductor whose inductance \(L(t)\) varies with time as a ferromagnetic core is pulled out of the coil. Which of the following differential equations correctly describes the current \(I(t)\) in the circuit while the core is being pulled?

![A single-loop circuit schematic oriented horizontally. On the left vertical segment is a battery labeled with constant emf \mathcal{E}, with its long positive line on top. On the top horizontal segment is a resistor labeled R. On the right vertical segment is an inductor coil labeled L(t). Inside the coil, a shaded rectangular cylinder representing a core is shown partially withdrawn, with a small arrow pointing upward labeled v indicating movement out of the coil. No other components or labels appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831218-WRPW7V.jpg)

- **A.** \(L(t) \dfrac{dI}{dt} + I R = \mathcal{E}\)
- **B.** \(L(t) \dfrac{dI}{dt} + I \left( R - \dfrac{dL}{dt} \right) = \mathcal{E}\)
- **C.** \(-L(t) \dfrac{dI}{dt} + I \left( R + \dfrac{dL}{dt} \right) = \mathcal{E}\)
- **D.** \(L(t) \dfrac{dI}{dt} + I \left( R + \dfrac{dL}{dt} \right) = \mathcal{E}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118673/*