---
title: "A thin straight wire segment of length \\(L\\) carrying a steady current \\(I\\) lies along the horizontal axis. Using the Biot–Savart law, the magnitude of the magnetic field at a point \\(P\\) located a perpendicular distance \\(d\\) from the midpoint of the segment is given by \\(B = \\dfrac{\\mu_0 I L}{2\\pi d \\sqrt{L^2 + 4d^2}}\\). Which of the following correctly describes the behavior of this magnetic field in the limits \\(L \\gg d\\) and \\(d \\gg L\\)?"
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url: "https://nerd-notes.com/ubq/121233/"
date_modified: "2026-08-23T04:58:58+00:00"
---

# A thin straight wire segment of length \(L\) carrying a steady current \(I\) lies along the horizontal axis. Using the Biot–Savart law, the magnitude of the magnetic field at a point \(P\) located a perpendicular distance \(d\) from the midpoint of the segment is given by \(B = \dfrac{\mu_0 I L}{2\pi d \sqrt{L^2 + 4d^2}}\). Which of the following correctly describes the behavior of this magnetic field in the limits \(L \gg d\) and \(d \gg L\)?

A thin straight wire segment of length \(L\) carrying a steady current \(I\) lies along the horizontal axis. Using the Biot–Savart law, the magnitude of the magnetic field at a point \(P\) located a perpendicular distance \(d\) from the midpoint of the segment is given by \(B = \dfrac{\mu_0 I L}{2\pi d \sqrt{L^2 + 4d^2}}\). Which of the following correctly describes the behavior of this magnetic field in the limits \(L \gg d\) and \(d \gg L\)?

![A horizontal line segment of length \(L\) centered at the origin, with an arrowhead at its right end indicating current \(I\) flowing to the right. A dashed vertical line of length \(d\) extends upward from the midpoint of the horizontal segment to a point marked with a solid dot labeled \(P\). A horizontal double-ended dimension arrow below the segment indicates the total length \(L\). A vertical double-ended dimension arrow to the left of the dashed line indicates the distance \(d\). No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1787461138-TbQvEM.jpg)

- **A.** For \(L \gg d\), \(B \to \dfrac{\mu_0 I}{2\pi d}\), consistent with Ampère's law for an infinitely long wire; for \(d \gg L\), \(B \approx \dfrac{\mu_0 I L}{4\pi d^2}\), consistent with the inverse-square decay of an isolated current element.
- **B.** For \(L \gg d\), \(B \to \dfrac{\mu_0 I}{4\pi d}\), consistent with the field near one end of a semi-infinite wire; for \(d \gg L\), \(B \approx \dfrac{\mu_0 I L}{4\pi d^2}\), consistent with the inverse-square decay of an isolated current element.
- **C.** For \(L \gg d\), \(B \to \dfrac{\mu_0 I}{2\pi d}\), consistent with Ampère's law for an infinitely long wire; for \(d \gg L\), \(B \approx \dfrac{\mu_0 I L^2}{4\pi d^3}\), consistent with the inverse-cube decay of a magnetic dipole.
- **D.** For \(L \gg d\), \(B \to \dfrac{\mu_0 I}{2\pi L}\), indicating decay governed by segment length rather than distance; for \(d \gg L\), \(B \approx \dfrac{\mu_0 I}{2\pi d}\), maintaining the cylindrical symmetry of an extended line.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/121233/*