---
title: "A rigid planar rectangular wire loop of width \\(w\\) and length \\(\\ell\\) lies in the \\(xy\\)-plane with its sides parallel to the coordinate axes. A time-independent, spatially non-uniform magnetic field perpendicular to the loop is described by \\(\\vec{B}(x, y, z) = \\beta x\\,\\hat{k}\\), where \\(\\beta\\) is a positive constant. The loop moves in the \\(+y\\)-direction with a constant velocity \\(\\vec{v} = v_0\\,\\hat{j}\\).  Which of the following statements correctly describes the net induced electromotive force (EMF) around the loop and provides the correct physical justification?"
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url: "https://nerd-notes.com/ubq/121295/"
date_modified: "2026-08-23T04:59:36+00:00"
---

# A rigid planar rectangular wire loop of width \(w\) and length \(\ell\) lies in the \(xy\)-plane with its sides parallel to the coordinate axes. A time-independent, spatially non-uniform magnetic field perpendicular to the loop is described by \(\vec{B}(x, y, z) = \beta x\,\hat{k}\), where \(\beta\) is a positive constant. The loop moves in the \(+y\)-direction with a constant velocity \(\vec{v} = v_0\,\hat{j}\).

Which of the following statements correctly describes the net induced electromotive force (EMF) around the loop and provides the correct physical justification?

A rigid planar rectangular wire loop of width \(w\) and length \(\ell\) lies in the \(xy\)-plane with its sides parallel to the coordinate axes. A time-independent, spatially non-uniform magnetic field perpendicular to the loop is described by \(\vec{B}(x, y, z) = \beta x\,\hat{k}\), where \(\beta\) is a positive constant. The loop moves in the \(+y\)-direction with a constant velocity \(\vec{v} = v_0\,\hat{j}\).

Which of the following statements correctly describes the net induced electromotive force (EMF) around the loop and provides the correct physical justification?

![A Cartesian coordinate system with a horizontal x-axis pointing right and a vertical y-axis pointing up. A rigid rectangular loop of width w along the x-axis and length \ell along the y-axis lies in the first quadrant of the xy-plane. A single vertical arrow labeled \vec{v} = v_0\hat{j} points upward from the center of the loop. Several out-of-page magnetic field symbols (circles with centered dots) are distributed across the region, with their spatial density increasing linearly from left to right along the x-axis, labeled \vec{B} = \beta x\hat{k}. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461175-2cF4U4.jpg)

- **A.** The induced EMF around the loop is nonzero because the magnetic field magnitude varies spatially across the width of the loop.
- **B.** The induced EMF around the loop is zero because the total magnetic flux enclosed by the loop does not change with time.
- **C.** The induced EMF around the loop is nonzero because magnetic forces on charge carriers in the top and bottom segments drive a net circulating current.
- **D.** The induced EMF around the loop is zero because the magnetic force on the charge carriers in every segment of the loop is identically zero.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121295/*