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title: "Three toroids, labeled 1, 2, and 3, each have \\(N\\) tightly wound turns carrying a current \\(I\\). All three toroids have inner radius \\(a\\) and outer radius \\(b\\). Toroid 1 has a rectangular cross-section of uniform height \\(h\\). Toroid 2 has a triangular cross-section whose height decreases linearly from \\(h\\) at \\(r = a\\) to \\(0\\) at \\(r = b\\). Toroid 3 has a triangular cross-section whose height increases linearly from \\(0\\) at \\(r = a\\) to \\(h\\) at \\(r = b\\). Which of the following correctly ranks the self-inductances \\(L_1\\), \\(L_2\\), and \\(L_3\\) of the three toroids?"
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url: "https://nerd-notes.com/ubq/118750/"
date_modified: "2026-08-04T08:14:07+00:00"
---

# Three toroids, labeled 1, 2, and 3, each have \(N\) tightly wound turns carrying a current \(I\). All three toroids have inner radius \(a\) and outer radius \(b\). Toroid 1 has a rectangular cross-section of uniform height \(h\). Toroid 2 has a triangular cross-section whose height decreases linearly from \(h\) at \(r = a\) to \(0\) at \(r = b\). Toroid 3 has a triangular cross-section whose height increases linearly from \(0\) at \(r = a\) to \(h\) at \(r = b\). Which of the following correctly ranks the self-inductances \(L_1\), \(L_2\), and \(L_3\) of the three toroids?

Three toroids, labeled 1, 2, and 3, each have \(N\) tightly wound turns carrying a current \(I\). All three toroids have inner radius \(a\) and outer radius \(b\). Toroid 1 has a rectangular cross-section of uniform height \(h\). Toroid 2 has a triangular cross-section whose height decreases linearly from \(h\) at \(r = a\) to \(0\) at \(r = b\). Toroid 3 has a triangular cross-section whose height increases linearly from \(0\) at \(r = a\) to \(h\) at \(r = b\). Which of the following correctly ranks the self-inductances \(L_1\), \(L_2\), and \(L_3\) of the three toroids?

![Three cross-sectional diagrams of toroids side by side, labeled 1, 2, and 3. Each diagram shows a central vertical axis of rotation and a cross-sectional area located between radial distance a and radial distance b from the central axis. In diagram 1, the cross-section is a rectangle of width b - a and uniform height h. In diagram 2, the cross-section is a right triangle with a vertical left side of height h at radial distance a, tapering down to a vertex at radial distance b. In diagram 3, the cross-section is a right triangle with a vertex at radial distance a, expanding up to a vertical right side of height h at radial distance b. A horizontal dashed baseline marks radial position a on the left and radial position b on the right for each diagram. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831247-oAOLC7.jpg)

- **A.** \(L_3 > L_2 > L_1\)
- **B.** \(L_1 > L_3 > L_2\)
- **C.** \(L_2 > L_3 > L_1\)
- **D.** \(L_1 > L_2 > L_3\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118750/*