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title: "Two identical circular coils, each of radius \\(R\\), are placed coaxially along the \\(z\\)-axis with their centers at \\(z = -s/2\\) and \\(z = +s/2\\), where the origin \\(z = 0\\) is defined at the midpoint between them. The coils carry equal constant currents in the same direction, and their separation is set to \\(s = R\\) to create a region of near-uniform magnetic field centered at the origin. Which of the following statements correctly describes the behavior of the net axial magnetic field \\(B(z)\\) near \\(z = 0\\) and provides the correct physical and mathematical justification?"
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url: "https://nerd-notes.com/ubq/124763/"
date_modified: "2026-09-28T14:11:07+00:00"
---

# Two identical circular coils, each of radius \(R\), are placed coaxially along the \(z\)-axis with their centers at \(z = -s/2\) and \(z = +s/2\), where the origin \(z = 0\) is defined at the midpoint between them. The coils carry equal constant currents in the same direction, and their separation is set to \(s = R\) to create a region of near-uniform magnetic field centered at the origin. Which of the following statements correctly describes the behavior of the net axial magnetic field \(B(z)\) near \(z = 0\) and provides the correct physical and mathematical justification?

Two identical circular coils, each of radius \(R\), are placed coaxially along the \(z\)-axis with their centers at \(z = -s/2\) and \(z = +s/2\), where the origin \(z = 0\) is defined at the midpoint between them. The coils carry equal constant currents in the same direction, and their separation is set to \(s = R\) to create a region of near-uniform magnetic field centered at the origin. Which of the following statements correctly describes the behavior of the net axial magnetic field \(B(z)\) near \(z = 0\) and provides the correct physical and mathematical justification?

![A schematic of two identical circular coils aligned coaxially along a horizontal central axis labeled z. The left coil of radius R is centered at z = -s/2, and the right coil of radius R is centered at z = +s/2, separated by horizontal distance s. The coils are rendered as vertically oriented ellipses to indicate three-dimensional perspective. On each coil, an arrow along the perimeter indicates current I circulating in the same direction. A horizontal dashed line represents the central axis passing through the centers of both coils, with a vertical tick mark at the midpoint labeled z = 0. A double-headed horizontal arrow below the coils indicates the separation labeled s. A vertical arrow extending from the center of the left coil to its upper edge is labeled R. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604667-utaruy.jpg)

- **A.** The net magnetic field is uniform near the midpoint because setting \(s = R\) makes the first derivative \(dB/dz\) equal to zero at \(z = 0\), whereas for other separations \(dB/dz \ne 0\) at the midpoint.
- **B.** The net magnetic field is uniform near the midpoint because the magnetic fields produced by the two coils point in opposite directions along the axis, causing their spatial variations to cancel.
- **C.** The net magnetic field is uniform near the midpoint because the first derivative \(dB/dz\) vanishes by symmetry for any separation, and setting \(s = R\) causes the second derivative \(d^2B/dz^2\) to also vanish at \(z = 0\).
- **D.** The net magnetic field is uniform near the midpoint because setting \(s = R\) maximizes the net magnetic field at \(z = 0\), which requires that the second derivative \(d^2B/dz^2\) be strictly negative.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124763/*