---
title: "A conducting rod of mass \\(m\\) and length \\(\\ell\\) is placed on two parallel, horizontal, frictionless conducting rails separated by a distance \\(\\ell\\) in a uniform vertical magnetic field of magnitude \\(B\\). The rails are connected at one end to an ideal capacitor of capacitance \\(C\\), and the electrical resistance of the rails and rod is negligible. Starting from rest at time \\(t = 0\\), a constant external horizontal force \\(F\\) pulls the rod along the rails directed away from the capacitor. Which of the following statements correctly describes the motion of the rod in the limit as \\(t \\to \\infty\\)?"
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url: "https://nerd-notes.com/ubq/125063/"
date_modified: "2026-09-28T14:14:13+00:00"
---

# A conducting rod of mass \(m\) and length \(\ell\) is placed on two parallel, horizontal, frictionless conducting rails separated by a distance \(\ell\) in a uniform vertical magnetic field of magnitude \(B\). The rails are connected at one end to an ideal capacitor of capacitance \(C\), and the electrical resistance of the rails and rod is negligible. Starting from rest at time \(t = 0\), a constant external horizontal force \(F\) pulls the rod along the rails directed away from the capacitor. Which of the following statements correctly describes the motion of the rod in the limit as \(t \to \infty\)?

A conducting rod of mass \(m\) and length \(\ell\) is placed on two parallel, horizontal, frictionless conducting rails separated by a distance \(\ell\) in a uniform vertical magnetic field of magnitude \(B\). The rails are connected at one end to an ideal capacitor of capacitance \(C\), and the electrical resistance of the rails and rod is negligible. Starting from rest at time \(t = 0\), a constant external horizontal force \(F\) pulls the rod along the rails directed away from the capacitor. Which of the following statements correctly describes the motion of the rod in the limit as \(t \to \infty\)?

![A top-down view of a horizontal rectangular rail system. Two parallel horizontal line segments represent conducting rails separated vertically by a distance labeled \(\ell\). At the left end, the two rails are connected to the two parallel plates of a capacitor labeled \(C\). A vertical line segment representing a conducting rod of mass \(m\) and length \(\ell\) rests perpendicularly across both rails. A horizontal arrow pointing to the right originates at the midpoint of the rod and is labeled \(F\). A uniform magnetic field is indicated by an array of exactly six evenly spaced \(\times\) symbols distributed between the rails, with a single label \(\vec{B}\) placed near the top right. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604853-XFpq7i.jpg)

- **A.** In the limit \(t \to \infty\), the rod's acceleration approaches zero and its speed approaches a constant terminal velocity, because the opposing magnetic force increases with speed until it balances \(F\).
- **B.** In the limit \(t \to \infty\), the rod's acceleration approaches \(\dfrac{F}{m}\), because the capacitor becomes fully charged and the induced current drops to zero.
- **C.** In the limit \(t \to \infty\), the rod's acceleration approaches zero, because all mechanical work done by the applied force is stored in the capacitor's electric field rather than increasing the rod's kinetic energy.
- **D.** In the limit \(t \to \infty\), the rod's acceleration remains constant at \(\dfrac{F}{m + C B^2 \ell^2}\) and its speed increases without bound, because the opposing magnetic force is proportional to acceleration rather than speed.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/125063/*