---
title: "A rigid rectangular loop of wire with width \\(w\\), length \\(L\\), and total resistance \\(R\\) lies in the \\(xy\\)-plane and moves in the \\(+x\\)-direction at a constant speed \\(v\\). A non-uniform magnetic field directed perpendicular to the \\(xy\\)-plane and out of the page (in the \\(+z\\)-direction) has magnitude \\(B(x) = \\alpha x\\), where \\(\\alpha\\) is a positive constant and \\(x > 0\\). The loop remains completely within the magnetic field at all times. Which of the following correctly pairs the direction of the induced current in the loop with the dependence of the magnitude of the required external pulling force on the loop’s position \\(x\\)?"
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url: "https://nerd-notes.com/ubq/125048/"
date_modified: "2026-09-28T14:13:39+00:00"
---

# A rigid rectangular loop of wire with width \(w\), length \(L\), and total resistance \(R\) lies in the \(xy\)-plane and moves in the \(+x\)-direction at a constant speed \(v\). A non-uniform magnetic field directed perpendicular to the \(xy\)-plane and out of the page (in the \(+z\)-direction) has magnitude \(B(x) = \alpha x\), where \(\alpha\) is a positive constant and \(x > 0\). The loop remains completely within the magnetic field at all times. Which of the following correctly pairs the direction of the induced current in the loop with the dependence of the magnitude of the required external pulling force on the loop’s position \(x\)?

A rigid rectangular loop of wire with width \(w\), length \(L\), and total resistance \(R\) lies in the \(xy\)-plane and moves in the \(+x\)-direction at a constant speed \(v\). A non-uniform magnetic field directed perpendicular to the \(xy\)-plane and out of the page (in the \(+z\)-direction) has magnitude \(B(x) = \alpha x\), where \(\alpha\) is a positive constant and \(x > 0\). The loop remains completely within the magnetic field at all times. Which of the following correctly pairs the direction of the induced current in the loop with the dependence of the magnitude of the required external pulling force on the loop's position \(x\)?

![A horizontal coordinate axis labeled x extends to the right, and a vertical coordinate axis labeled y extends upward, meeting at an origin labeled O. In the first quadrant, a thin rectangular wire loop of width w along the x-direction and height L along the y-direction lies in the xy-plane. The left edge of the loop is positioned at horizontal coordinate x, and the right edge is positioned at x + w. A horizontal arrow pointing to the right labeled \vec{v} is attached to the center of the right edge of the loop. Across the region, a grid of circle symbols with central dots indicates a magnetic field directed out of the page. The horizontal spacing between adjacent dots decreases linearly from left to right to indicate increasing field strength. A text label near the top right indicates B(x) = \alpha x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604818-XLvirE.jpg)

- **A.** Direction of current: Clockwise; Dependence of force magnitude: Independent of \(x\)
- **B.** Direction of current: Clockwise; Dependence of force magnitude: Directly proportional to \(x\)
- **C.** Direction of current: Counterclockwise; Dependence of force magnitude: Independent of \(x\)
- **D.** Direction of current: Counterclockwise; Dependence of force magnitude: Directly proportional to \(x\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/125048/*