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title: "A toroidal inductor with a rectangular cross section has height \\(h\\), inner radius \\(a\\), outer radius \\(b\\), and \\(N\\) tightly wound turns. Integrating the magnetic field across the core yields the self-inductance \\(L = \\dfrac{\\mu_0 N^2 h}{2\\pi} \\ln\\left(\\dfrac{b}{a}\\right)\\). Let \\(\\delta = b – a\\) denote the radial thickness of the toroid. In the limiting case of a very slender toroid where \\(\\delta \\ll a\\), which of the following expressions correctly approximates \\(L\\), and what is its physical interpretation?"
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date_modified: "2026-08-23T04:59:44+00:00"
---

# A toroidal inductor with a rectangular cross section has height \(h\), inner radius \(a\), outer radius \(b\), and \(N\) tightly wound turns. Integrating the magnetic field across the core yields the self-inductance \(L = \dfrac{\mu_0 N^2 h}{2\pi} \ln\left(\dfrac{b}{a}\right)\). Let \(\delta = b – a\) denote the radial thickness of the toroid. In the limiting case of a very slender toroid where \(\delta \ll a\), which of the following expressions correctly approximates \(L\), and what is its physical interpretation?

A toroidal inductor with a rectangular cross section has height \(h\), inner radius \(a\), outer radius \(b\), and \(N\) tightly wound turns. Integrating the magnetic field across the core yields the self-inductance \(L = \dfrac{\mu_0 N^2 h}{2\pi} \ln\left(\dfrac{b}{a}\right)\). Let \(\delta = b - a\) denote the radial thickness of the toroid. In the limiting case of a very slender toroid where \(\delta \ll a\), which of the following expressions correctly approximates \(L\), and what is its physical interpretation?

![A cross-sectional perspective view of a toroidal inductor with a rectangular cross section. A central vertical dashed axis of symmetry passes through the center. A horizontal line segment extends from the central axis to the inner boundary labeled a. A second horizontal line segment extends from the central axis to the outer boundary labeled b. The rectangular cross section has a vertical dimension labeled h and a horizontal radial width labeled \delta = b - a. Closely spaced wire turns wrap around the core. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461184-4VBlRP.jpg)

- **A.** \(L \approx \dfrac{\mu_0 N^2 h}{2\pi} \left(\dfrac{a}{\delta}\right)\), because the magnetic field diverges at the inner radius as the cross section becomes infinitely thin.
- **B.** \(L \approx \dfrac{\mu_0 N^2 h\delta}{2\pi a}\), which is identical to the self-inductance of an ideal solenoid of cross-sectional area \(A = h\delta\) and length \(\ell = 2\pi a\).
- **C.** \(L \approx \dfrac{\mu_0 N^2 h\delta}{4\pi a}\), which accounts for the arithmetic average of the magnetic field between the inner and outer boundaries.
- **D.** \(L \approx \dfrac{\mu_0 N^2 h\delta^2}{2\pi a^2}\), because the curvature of the toroid suppresses magnetic flux to second order in the radial thickness.

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