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title: "A thin, rigid wire carries a steady current \\(I\\) in the \\(xy\\)-plane along the parabolic path \\(y = kx^2\\) from the origin \\((0,0)\\) to the point \\((L, kL^2)\\), where \\(k\\) and \\(L\\) are positive constants. The current flows along the wire in the direction of increasing \\(x\\). A non-uniform external magnetic field perpendicular to the \\(xy\\)-plane is given by \\(\\vec{B}(x) = B_0 \\left(\\dfrac{x}{L}\\right) \\hat{k}\\), where \\(B_0\\) is a positive constant and \\(\\hat{k}\\) is the unit vector pointing in the \\(+z\\)-direction. Which of the following integral expressions correctly represents the net magnetic force vector \\(\\vec{F}\\) exerted on the wire?"
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url: "https://nerd-notes.com/ubq/121257/"
date_modified: "2026-08-23T04:59:03+00:00"
---

# A thin, rigid wire carries a steady current \(I\) in the \(xy\)-plane along the parabolic path \(y = kx^2\) from the origin \((0,0)\) to the point \((L, kL^2)\), where \(k\) and \(L\) are positive constants. The current flows along the wire in the direction of increasing \(x\). A non-uniform external magnetic field perpendicular to the \(xy\)-plane is given by \(\vec{B}(x) = B_0 \left(\dfrac{x}{L}\right) \hat{k}\), where \(B_0\) is a positive constant and \(\hat{k}\) is the unit vector pointing in the \(+z\)-direction. Which of the following integral expressions correctly represents the net magnetic force vector \(\vec{F}\) exerted on the wire?

A thin, rigid wire carries a steady current \(I\) in the \(xy\)-plane along the parabolic path \(y = kx^2\) from the origin \((0,0)\) to the point \((L, kL^2)\), where \(k\) and \(L\) are positive constants. The current flows along the wire in the direction of increasing \(x\). A non-uniform external magnetic field perpendicular to the \(xy\)-plane is given by \(\vec{B}(x) = B_0 \left(\dfrac{x}{L}\right) \hat{k}\), where \(B_0\) is a positive constant and \(\hat{k}\) is the unit vector pointing in the \(+z\)-direction. Which of the following integral expressions correctly represents the net magnetic force vector \(\vec{F}\) exerted on the wire?

![Cartesian coordinate axes x (horizontal, pointing right) and y (vertical, pointing up) intersecting at the origin labeled O. A smooth solid parabolic curve lies in the first quadrant, starting at O (0,0) and terminating at a point labeled (L, kL^2). An arrowhead on the curve points along the path toward the upper-right, labeled with current I. A vertical dashed line extends from (L, kL^2) down to the horizontal axis at tick mark L, and a horizontal dashed line extends from (L, kL^2) left to the vertical axis at tick mark kL^2. In the region 0 to L along the horizontal axis, exactly six out-of-page magnetic field symbols (circles with centered dots) are arranged in two columns of three, labeled B(x) = B_0 (x/L) k-hat, with the right column spaced more closely to indicate increasing field strength with x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461143-ZN1Znx.jpg)

- **A.** \(\dfrac{I B_0}{L} \int_0^L \left( x\,\hat{i} - 2k x^2\,\hat{j} \right) dx\)
- **B.** \(\dfrac{I B_0}{L} \int_0^L \left( 2k x^2\,\hat{i} - x\,\hat{j} \right) dx\)
- **C.** \(\dfrac{I B_0}{L} \int_0^L \left( 2k x\,\hat{i} - \hat{j} \right) \sqrt{1 + 4k^2 x^2}\,dx\)
- **D.** \(\dfrac{I B_0}{L} \int_0^L \left( k x^2\,\hat{i} - x\,\hat{j} \right) dx\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121257/*