---
title: "A small magnetic dipole of moment \\(\\vec{m} = m\\hat{z}\\) is fixed at the origin. A thin circular ring of radius \\(R\\) carrying a steady counterclockwise current \\(I\\) (as viewed from above) lies in a horizontal plane centered at \\((0, 0, z)\\) on the positive \\(z\\)-axis, where \\(z > 0\\). The net vertical magnetic force exerted on the ring by the dipole is given by  \\[ F_z = -\\dfrac{3\\mu_0 m I z R^2}{2(z^2 + R^2)^{5/2}} \\]  In the limit where the loop radius is much smaller than the distance to the dipole (\\(R \\ll z\\)), which of the following expressions correctly approximates \\(F_z\\), and what physical mechanism explains its dependence on \\(R\\)?"
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url: "https://nerd-notes.com/ubq/124921/"
date_modified: "2026-09-28T14:12:01+00:00"
---

# A small magnetic dipole of moment \(\vec{m} = m\hat{z}\) is fixed at the origin. A thin circular ring of radius \(R\) carrying a steady counterclockwise current \(I\) (as viewed from above) lies in a horizontal plane centered at \((0, 0, z)\) on the positive \(z\)-axis, where \(z > 0\). The net vertical magnetic force exerted on the ring by the dipole is given by

\[ F_z = -\dfrac{3\mu_0 m I z R^2}{2(z^2 + R^2)^{5/2}} \]

In the limit where the loop radius is much smaller than the distance to the dipole (\(R \ll z\)), which of the following expressions correctly approximates \(F_z\), and what physical mechanism explains its dependence on \(R\)?

A small magnetic dipole of moment \(\vec{m} = m\hat{z}\) is fixed at the origin. A thin circular ring of radius \(R\) carrying a steady counterclockwise current \(I\) (as viewed from above) lies in a horizontal plane centered at \((0, 0, z)\) on the positive \(z\)-axis, where \(z > 0\). The net vertical magnetic force exerted on the ring by the dipole is given by

\[ F_z = -\dfrac{3\mu_0 m I z R^2}{2(z^2 + R^2)^{5/2}} \]

In the limit where the loop radius is much smaller than the distance to the dipole (\(R \ll z\)), which of the following expressions correctly approximates \(F_z\), and what physical mechanism explains its dependence on \(R\)?

![A vertical coordinate axis labeled z extends upward through the center of the diagram. At the origin on the z-axis, a short vertical arrow pointing in the positive z-direction is labeled m. A horizontal dashed line indicates the plane at the origin. Located at a distance z above the origin along the vertical axis is a horizontal ellipse representing a circular ring of radius R. A dashed line segment extends horizontally from the center of the ellipse to its right edge, labeled R. An arrow along the front curved edge of the ellipse points to the right to indicate a counterclockwise current labeled I. A vertical double-headed dimension arrow extends along the axis between the origin and the center of the ring, labeled z. Exactly four smooth curved lines representing magnetic field lines emerge from the origin, curving symmetrically outward and upward to pass through and around the ring. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604721-j4dktr.jpg)

- **A.** \(F_z \approx -\dfrac{3\mu_0 m I R}{2z^3}\), because the loop circumference is proportional to \(R\) and the magnetic field magnitude across the ring is approximately constant.
- **B.** \(F_z \approx -\dfrac{3\mu_0 m I R^2}{2z^4}\), because the loop circumference is proportional to \(R\) and the radial magnetic field component \(B_r\) that produces the vertical force is also proportional to \(R\).
- **C.** \(F_z \approx -\dfrac{3\mu_0 m I R^2}{2z^4}\), because the area enclosed by the loop is proportional to \(R^2\) and the vertical magnetic force is produced directly by the axial magnetic field component \(B_z\).
- **D.** \(F_z \approx -\dfrac{3\mu_0 m I}{2z^2}\), because as the loop radius approaches zero, the loop shrinks to a point on the axis and experiences the magnetic field gradient independently of its dimensions.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124921/*