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title: "A thin, rigid wire bent into a semicircle of radius \\(R\\) lies in the \\(xy\\)-plane with its center of curvature at the origin, occupying the region \\(y \\ge 0\\). The wire carries a steady current \\(I\\) directed counterclockwise from \\((R, 0, 0)\\) to \\((-R, 0, 0)\\). A uniform magnetic field \\(\\vec{B}\\) of magnitude \\(B_0\\) lies in the \\(yz\\)-plane and makes a constant angle \\(\\phi\\) with the positive \\(y\\)-axis, such that \\(\\vec{B} = B_0\\cos\\phi\\,\\hat{j} + B_0\\sin\\phi\\,\\hat{k}\\), where \\(0 < \\phi < \\dfrac{\\pi}{2}\\). Which of the following expressions represents a correct integral setup to determine the \\(z\\)-component of the net magnetic force, \\(F_z\\), exerted on the wire?"
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url: "https://nerd-notes.com/ubq/124849/"
date_modified: "2026-09-28T14:11:39+00:00"
---

# A thin, rigid wire bent into a semicircle of radius \(R\) lies in the \(xy\)-plane with its center of curvature at the origin, occupying the region \(y \ge 0\). The wire carries a steady current \(I\) directed counterclockwise from \((R, 0, 0)\) to \((-R, 0, 0)\). A uniform magnetic field \(\vec{B}\) of magnitude \(B_0\) lies in the \(yz\)-plane and makes a constant angle \(\phi\) with the positive \(y\)-axis, such that \(\vec{B} = B_0\cos\phi\,\hat{j} + B_0\sin\phi\,\hat{k}\), where \(0 < \phi < \dfrac{\pi}{2}\). Which of the following expressions represents a correct integral setup to determine the \(z\)-component of the net magnetic force, \(F_z\), exerted on the wire?

A thin, rigid wire bent into a semicircle of radius \(R\) lies in the \(xy\)-plane with its center of curvature at the origin, occupying the region \(y \ge 0\). The wire carries a steady current \(I\) directed counterclockwise from \((R, 0, 0)\) to \((-R, 0, 0)\). A uniform magnetic field \(\vec{B}\) of magnitude \(B_0\) lies in the \(yz\)-plane and makes a constant angle \(\phi\) with the positive \(y\)-axis, such that \(\vec{B} = B_0\cos\phi\,\hat{j} + B_0\sin\phi\,\hat{k}\), where \(0 < \phi < \dfrac{\pi}{2}\). Which of the following expressions represents a correct integral setup to determine the \(z\)-component of the net magnetic force, \(F_z\), exerted on the wire?

![A three-dimensional Cartesian coordinate system with three mutually perpendicular axes labeled \(x\), \(y\), and \(z\). The horizontal axis pointing to the right is labeled \(x\), the vertical axis pointing upward is labeled \(z\), and the axis pointing obliquely forward and downward to the left is labeled \(y\). A semicircular arc of radius \(R\) lies in the \(xy\)-plane with its center at the origin, curving through positive \(y\) from the point \((R,0,0)\) on the positive \(x\)-axis to the point \((-R,0,0)\) on the negative \(x\)-axis. An arrow along the arc indicates current \(I\) directed counterclockwise from \((R,0,0)\) toward \((-R,0,0)\). An angle arc labeled \(\theta\) is drawn in the \(xy\)-plane from the positive \(x\)-axis counterclockwise to a point on the semicircle. A straight vector arrow representing the uniform magnetic field \(\vec{B}\) originates at the origin, lies in the \(yz\)-plane, and points into the region where \(y > 0\) and \(z > 0\), with an angle arc labeled \(\phi\) between the vector and the positive \(y\)-axis. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604699-ygtJcN.jpg)

- **A.** \(I B_0 R \cos\phi \int_0^{\pi} \cos\theta \, d\theta\)
- **B.** \(-I B_0 R \sin\phi \int_0^{\pi} \sin\theta \, d\theta\)
- **C.** \(I B_0 R \cos\phi \int_0^{\pi} \sin\theta \, d\theta\)
- **D.** \(-I B_0 R \cos\phi \int_0^{\pi} \sin\theta \, d\theta\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124849/*