---
title: "A square loop of wire with side length \\(L\\) lies in a region containing a non-uniform magnetic field \\(\\vec{B} = B_0 \\left(\\dfrac{x}{L}\\right) \\hat{k}\\), where \\(B_0\\) is a positive constant and \\(x \\ge 0\\). One edge of the loop is fixed along the \\(y\\)-axis from \\(y = 0\\) to \\(y = L\\). The loop is tilted about the \\(y\\)-axis by an angle \\(\\theta\\) relative to the \\(xy\\)-plane. What is the ratio of the magnetic flux through the loop when \\(\\theta = 60^\\circ\\) to the magnetic flux through the loop when \\(\\theta = 0^\\circ\\)?"
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/118767/"
date_modified: "2026-08-04T08:14:29+00:00"
---

# A square loop of wire with side length \(L\) lies in a region containing a non-uniform magnetic field \(\vec{B} = B_0 \left(\dfrac{x}{L}\right) \hat{k}\), where \(B_0\) is a positive constant and \(x \ge 0\). One edge of the loop is fixed along the \(y\)-axis from \(y = 0\) to \(y = L\). The loop is tilted about the \(y\)-axis by an angle \(\theta\) relative to the \(xy\)-plane. What is the ratio of the magnetic flux through the loop when \(\theta = 60^\circ\) to the magnetic flux through the loop when \(\theta = 0^\circ\)?

A square loop of wire with side length \(L\) lies in a region containing a non-uniform magnetic field \(\vec{B} = B_0 \left(\dfrac{x}{L}\right) \hat{k}\), where \(B_0\) is a positive constant and \(x \ge 0\). One edge of the loop is fixed along the \(y\)-axis from \(y = 0\) to \(y = L\). The loop is tilted about the \(y\)-axis by an angle \(\theta\) relative to the \(xy\)-plane. What is the ratio of the magnetic flux through the loop when \(\theta = 60^\circ\) to the magnetic flux through the loop when \(\theta = 0^\circ\)?

![A 3D Cartesian coordinate system with x, y, and z axes labeled at the ends of thin black lines with arrowheads. A flat square loop of side length L is situated in the xy-plane with one edge lying along the y-axis from y = 0 to y = L. A second identical square loop shares the same edge along the y-axis but is tilted upward into the xz-plane by an angle \theta relative to the xy-plane. An arc labeled \theta indicates the tilt angle between the two loop planes. Vertical upward arrows parallel to the z-axis represent a magnetic field \vec{B}; the arrows increase in length at larger values of x. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831269-Bjw4hk.jpg)

- **A.** \(\dfrac{1}{8}\)
- **B.** \(\dfrac{1}{4}\)
- **C.** \(\dfrac{1}{2}\)
- **D.** \(\dfrac{3}{4}\)

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118767/*