---
title: "Students measure the magnetic field magnitude \\(B\\) along the central axis of a circular wire loop of radius \\(R\\) carrying a steady current \\(I\\), as a function of the axial distance \\(z\\) from the center of the loop.  According to the Biot-Savart law, the on-axis magnetic field is given by \\[ B(z) = \\dfrac{\\mu_0 I R^2}{2(R^2 + z^2)^{3/2}} \\]  The students wish to construct a single linear graph using all their data points (\\(z \\ge 0\\)) to confirm this functional relationship, which approaches the inverse-cube dipole dependence \\(B \\propto z^{-3}\\) for \\(z \\gg R\\). Which of the following pairs of quantities should be plotted on the vertical and horizontal axes to produce a linear graph, and what relationship correctly yields the radius \\(R\\) from the vertical intercept \\(b\\) and slope \\(m\\) of the best-fit line?"
description: "## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please […]"
url: "https://nerd-notes.com/ubq/124960/"
date_modified: "2026-09-28T14:12:14+00:00"
---

# Students measure the magnetic field magnitude \(B\) along the central axis of a circular wire loop of radius \(R\) carrying a steady current \(I\), as a function of the axial distance \(z\) from the center of the loop.

According to the Biot-Savart law, the on-axis magnetic field is given by
\[ B(z) = \dfrac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}} \]

The students wish to construct a single linear graph using all their data points (\(z \ge 0\)) to confirm this functional relationship, which approaches the inverse-cube dipole dependence \(B \propto z^{-3}\) for \(z \gg R\). Which of the following pairs of quantities should be plotted on the vertical and horizontal axes to produce a linear graph, and what relationship correctly yields the radius \(R\) from the vertical intercept \(b\) and slope \(m\) of the best-fit line?

Students measure the magnetic field magnitude \(B\) along the central axis of a circular wire loop of radius \(R\) carrying a steady current \(I\), as a function of the axial distance \(z\) from the center of the loop.

According to the Biot-Savart law, the on-axis magnetic field is given by
\[ B(z) = \dfrac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}} \]

The students wish to construct a single linear graph using all their data points (\(z \ge 0\)) to confirm this functional relationship, which approaches the inverse-cube dipole dependence \(B \propto z^{-3}\) for \(z \gg R\). Which of the following pairs of quantities should be plotted on the vertical and horizontal axes to produce a linear graph, and what relationship correctly yields the radius \(R\) from the vertical intercept \(b\) and slope \(m\) of the best-fit line?

![A flat circular wire loop is drawn as an ellipse in perspective, positioned horizontally in the lower half of the frame. A curved arrow on the loop indicates counterclockwise current labeled I when viewed from above. A straight line segment extends from the center of the loop toward the upper-right edge of the loop, labeled R. A vertical dashed line passes through the geometric center of the loop, extending upward to represent the central axis, labeled z at the top. A distinct dot marks an axial point on this vertical line above the center; a double-headed vertical dimension line between the loop center and the dot is labeled z. A solid straight arrow begins at the dot and points straight upward along the dashed vertical line, labeled \vec{B}. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1790604734-Ui8Hcj.jpg)

- **A.** Vertical axis: \(B^{-3/2}\); Horizontal axis: \(z^2\); Radius: \(R = \sqrt{\dfrac{b}{m}}\)
- **B.** Vertical axis: \(B^{-2/3}\); Horizontal axis: \(z^2\); Radius: \(R = \sqrt{\dfrac{b}{m}}\)
- **C.** Vertical axis: \(B^{-2/3}\); Horizontal axis: \(z\); Radius: \(R = \dfrac{b}{m}\)
- **D.** Vertical axis: \(B^{-2/3}\); Horizontal axis: \(z^2\); Radius: \(R = \sqrt{\dfrac{m}{b}}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/124960/*