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title: "A straight wire of length \\(L\\) lies along the \\(x\\)-axis from \\(x = 0\\) to \\(x = L\\) and carries a constant current \\(I\\) in the \\(+x\\)-direction. The wire sits in a non-uniform magnetic field directed in the \\(+z\\)-direction whose magnitude increases linearly with position according to \\(B(x) = B_0 + Cx\\), where \\(B_0\\) and \\(C\\) are positive constants. The total magnetic force on the wire is given by \\(F = I L B_{\\text{avg}}\\), where \\(B_{\\text{avg}} = B(L/2)\\) is the field magnitude at the midpoint of the wire. Which of the following provides the correct physical and mathematical explanation for why the net force depends on \\(B(L/2)\\)?"
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url: "https://nerd-notes.com/ubq/118518/"
date_modified: "2026-08-04T08:11:13+00:00"
---

# A straight wire of length \(L\) lies along the \(x\)-axis from \(x = 0\) to \(x = L\) and carries a constant current \(I\) in the \(+x\)-direction. The wire sits in a non-uniform magnetic field directed in the \(+z\)-direction whose magnitude increases linearly with position according to \(B(x) = B_0 + Cx\), where \(B_0\) and \(C\) are positive constants. The total magnetic force on the wire is given by \(F = I L B_{\text{avg}}\), where \(B_{\text{avg}} = B(L/2)\) is the field magnitude at the midpoint of the wire. Which of the following provides the correct physical and mathematical explanation for why the net force depends on \(B(L/2)\)?

A straight wire of length \(L\) lies along the \(x\)-axis from \(x = 0\) to \(x = L\) and carries a constant current \(I\) in the \(+x\)-direction. The wire sits in a non-uniform magnetic field directed in the \(+z\)-direction whose magnitude increases linearly with position according to \(B(x) = B_0 + Cx\), where \(B_0\) and \(C\) are positive constants. The total magnetic force on the wire is given by \(F = I L B_{\text{avg}}\), where \(B_{\text{avg}} = B(L/2)\) is the field magnitude at the midpoint of the wire. Which of the following provides the correct physical and mathematical explanation for why the net force depends on \(B(L/2)\)?

![A horizontal axis labeled x contains a straight thick line segment representing a wire extending from a tick mark labeled x = 0 to a tick mark labeled x = L. A tick mark at the center of the wire is labeled x = L/2. A black arrow above the wire points to the right and is labeled I. Above the wire, three circled dot symbols representing magnetic field vectors pointing out of the page are spaced at equal intervals: the smallest symbol is above x = 0 labeled B_0, a medium symbol is above x = L/2 labeled B(L/2), and the largest symbol is above x = L labeled B(L). A horizontal arrow below the x-axis indicates increasing x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831072-c7xKZp.jpg)

- **A.** The magnetic force acts on the wire at its center of mass, so only the magnitude of the magnetic field at \(x = L/2\) contributes to the total force on the current.
- **B.** Summing the differential forces \(dF = I B(x)\,dx\) along the wire requires integrating \(B(x)\) over the interval, and for any linearly varying function, the average value over the interval equals the function evaluated at the midpoint \(x = L/2\).
- **C.** The magnetic forces on the segments to the left and right of \(x = L/2\) act in opposite directions, canceling each other out and leaving a net force proportional to \(B(L/2)\).
- **D.** The linear increase in magnetic field strength induces a back-EMF that causes the current \(I\) to decrease linearly along the wire, keeping the product \(I B(x)\) constant and equal to \(I B(L/2)\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118518/*