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title: "A student designs an electromagnetic system consisting of a closely wound toroid with inner radius \\(a\\), outer radius \\(b\\), and \\(N\\) tightly wound turns carrying a steady current \\(I\\). A magnetic field sensor is mounted at the geometric center of the toroid, located on the central axis of symmetry in the plane of the coils (\\(r = 0\\)). The student then replaces the toroid with a modified version having \\(2N\\) turns and an inner radius of \\(a/2\\) (with outer radius \\(b > a/2\\)), while maintaining the same current \\(I\\). By what factor does the magnitude of the magnetic field detected by the sensor change?"
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url: "https://nerd-notes.com/ubq/125008/"
date_modified: "2026-09-28T14:13:12+00:00"
---

# A student designs an electromagnetic system consisting of a closely wound toroid with inner radius \(a\), outer radius \(b\), and \(N\) tightly wound turns carrying a steady current \(I\). A magnetic field sensor is mounted at the geometric center of the toroid, located on the central axis of symmetry in the plane of the coils (\(r = 0\)). The student then replaces the toroid with a modified version having \(2N\) turns and an inner radius of \(a/2\) (with outer radius \(b > a/2\)), while maintaining the same current \(I\). By what factor does the magnitude of the magnetic field detected by the sensor change?

A student designs an electromagnetic system consisting of a closely wound toroid with inner radius \(a\), outer radius \(b\), and \(N\) tightly wound turns carrying a steady current \(I\). A magnetic field sensor is mounted at the geometric center of the toroid, located on the central axis of symmetry in the plane of the coils (\(r = 0\)). The student then replaces the toroid with a modified version having \(2N\) turns and an inner radius of \(a/2\) (with outer radius \(b > a/2\)), while maintaining the same current \(I\). By what factor does the magnitude of the magnetic field detected by the sensor change?

![A top-down view of a toroidal coil centered at the origin. Two concentric dashed circles represent the inner boundary of radius \(a\) and the outer boundary of radius \(b\). In the annular region between the two circles, closely spaced radial line segments represent the tightly wound turns of wire. A solid dot at the geometric center of the inner circle is labeled with a crosshair, marked with a small square labeled sensor at \(r = 0\). A straight radial dashed line extends from the center to the inner circular boundary with an arrow labeled \(a\), and a second straight radial dashed line extends from the center to the outer circular boundary with an arrow labeled \(b\). A curved arrow along one of the turns indicates current \(I\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604792-nrNOaD.jpg)

- **A.** It decreases by a factor of \(2\).
- **B.** It increases by a factor of \(2\).
- **C.** It increases by a factor of \(4\).
- **D.** The magnetic field is zero in both configurations.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/125008/*