---
title: "A closed surface \\(S\\) is situated in the non-uniform magnetic field produced by a nearby magnetic dipole, as shown. Magnetic field lines pass through various regions of the surface, yet the net magnetic flux \\(\\Phi_B = \\oint_S \\vec{B} \\cdot d\\vec{A}\\) through the closed surface is equal to zero. Which of the following statements correctly explains why the net magnetic flux through the closed surface is zero?"
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url: "https://nerd-notes.com/ubq/118671/"
date_modified: "2026-08-04T08:13:37+00:00"
---

# A closed surface \(S\) is situated in the non-uniform magnetic field produced by a nearby magnetic dipole, as shown. Magnetic field lines pass through various regions of the surface, yet the net magnetic flux \(\Phi_B = \oint_S \vec{B} \cdot d\vec{A}\) through the closed surface is equal to zero. Which of the following statements correctly explains why the net magnetic flux through the closed surface is zero?

A closed surface \(S\) is situated in the non-uniform magnetic field produced by a nearby magnetic dipole, as shown. Magnetic field lines pass through various regions of the surface, yet the net magnetic flux \(\Phi_B = \oint_S \vec{B} \cdot d\vec{A}\) through the closed surface is equal to zero. Which of the following statements correctly explains why the net magnetic flux through the closed surface is zero?

![A horizontal bar magnet is shown on the left with its North pole labeled N facing right and South pole labeled S facing left. To the right of the bar magnet, a smooth, closed 3D oval surface labeled S is drawn in space. Magnetic field lines emerge from the N pole, curve toward the right, penetrate the left side of surface S (entering the enclosed volume), pass through the interior region, and penetrate the right side of surface S (exiting the volume), continuing in continuous curves back toward the S pole. Two outward-pointing surface normal vector arrows labeled d\vec{A} are shown on surface S: one on the left pointing outward to the left, and one on the right pointing outward to the right. No other text, labels, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785831217-sWkjk6.jpg)

- **A.** The magnetic field within the volume enclosed by surface \(S\) is identically zero because the surface shields the interior from external magnetic fields.
- **B.** The magnetic field vectors at points on opposite sides of surface \(S\) have equal magnitudes and opposite directions, causing their scalar products with area vectors to cancel.
- **C.** Magnetic field lines form continuous closed loops without sources or sinks, so any field line that enters surface \(S\) must also exit surface \(S\).
- **D.** The magnetic field vector \(\vec{B}\) is everywhere perpendicular to the outward surface normal vector \(d\vec{A}\), making the integrand \(\vec{B} \cdot d\vec{A}\) zero at every point on surface \(S\).

*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/118671/*