---
title: "An infinite nonconducting flat sheet lying in the xy-plane carries a uniform surface current density \\(K\\) in the \\(+x\\)-direction. Which of the following provides the correct explanation using symmetry and Ampère’s law for why the magnitude of the resulting magnetic field is uniform and independent of the distance from the sheet?"
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url: "https://nerd-notes.com/ubq/118593/"
date_modified: "2026-08-04T08:11:33+00:00"
---

# An infinite nonconducting flat sheet lying in the xy-plane carries a uniform surface current density \(K\) in the \(+x\)-direction. Which of the following provides the correct explanation using symmetry and Ampère’s law for why the magnitude of the resulting magnetic field is uniform and independent of the distance from the sheet?

An infinite nonconducting flat sheet lying in the xy-plane carries a uniform surface current density \(K\) in the \(+x\)-direction. Which of the following provides the correct explanation using symmetry and Ampère's law for why the magnitude of the resulting magnetic field is uniform and independent of the distance from the sheet?

![A perspective view of an infinite flat sheet lying parallel to the xy-plane, carrying a uniform surface current density K directed in the +x direction, indicated by several parallel horizontal arrows labeled K on the sheet surface. A rectangular Amperian loop of width w and height 2z straddles the sheet vertically, extending a distance z above and z below the sheet. The top segment of the loop of length w is parallel to the y-axis, located at height z above the sheet, with an arrow indicating a magnetic field vector B pointing in the -y direction. The bottom segment of length w is located at distance z below the sheet, with an arrow indicating a magnetic field vector B pointing in the +y direction. The two vertical sides of the rectangular loop of length 2z pass perpendicularly through the sheet. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831092-S4yLtk.jpg)

- **A.** The magnetic field contribution from an individual current filament decreases inversely with distance, but integrating over the infinite sheet causes the total field to scale logarithmically with distance, which approaches a constant value at macroscopic scales.
- **B.** By Gauss's law for magnetism, the net magnetic flux through a rectangular Gaussian pillbox straddling the sheet must equal zero, forcing the magnetic field magnitudes on the top and bottom faces to be equal and independent of distance.
- **C.** An Amperian loop enclosing a section of the sheet experiences a line integral \(\oint \vec{B} \cdot d\vec{\ell}\) dominated by the vertical segments perpendicular to the sheet, whose lengths increase linearly with distance to balance the enclosed current.
- **D.** Symmetry dictates that the magnetic field is parallel to the sheet and uniform in magnitude at any fixed distance; choosing a rectangular Amperian loop perpendicular to the current yields zero line integral along the sides perpendicular to the sheet, so the total enclosed current depends only on the loop width, leaving the field magnitude independent of the height of the loop.

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118593/*