5.9.2. Gauss's Law for Magnetism

Table of Contents

Magnetic flux through a closed surface

Gauss's law for magnetism is a statement about magnetic field lines and closed surfaces. It says that the total magnetic flux through any closed surface is zero.

In words, as many magnetic field lines enter a closed surface as leave it. There is no net magnetic flow outward or inward through a closed boundary.

Mathematically, the law is written as

$$
\oint \vec{B} \cdot d\vec{A} = 0
$$

Here, $\vec{B}$ is the magnetic field, and $d\vec{A}$ is a small area element pointing outward from the closed surface. The circle on the integral sign means that the integration is over a closed surface.

Gauss's law for magnetism:
$$
\oint \vec{B} \cdot d\vec{A} = 0
$$
The net magnetic flux through every closed surface is zero.

Physical meaning

This law tells us something fundamental about magnetism. Electric charges can exist alone, positive or negative, but magnetic poles do not appear isolated in ordinary physics. If you cut a bar magnet in half, you do not get one piece with only a north pole and another with only a south pole. Instead, each piece becomes a smaller magnet with both a north and a south pole.

Because of this, magnetic field lines do not begin or end at isolated magnetic charges. Instead, they form continuous loops. A field line that leaves one part of a magnet must return elsewhere.

This is why the total flux through a closed surface must vanish. If some magnetic field lines leave the surface, the same number must re-enter somewhere else.

Comparison with electric flux

Gauss's law for electricity relates electric flux through a closed surface to the electric charge enclosed. For magnetism, there is no corresponding enclosed magnetic charge in standard classical electromagnetism. So the magnetic version gives zero for every closed surface.

The contrast is shown below.

Law	Mathematical form	Source inside closed surface
Gauss's law for electricity	$\oint \vec{E}\cdot d\vec{A} = \dfrac{Q_{\text{enc}}}{\varepsilon_0}$	Electric charge
Gauss's law for magnetism	$\oint \vec{B}\cdot d\vec{A} = 0$	No magnetic monopole

Field line picture

A useful visual idea is that magnetic field lines are closed loops. Around a bar magnet, lines emerge from the north side and enter the south side outside the magnet, then continue through the magnet itself to complete the loop.

Rendering scientific sketch…

Magnetic field lines form closed loops

If we imagine a closed surface around the magnet, some field lines pass outward through one part of the surface and inward through another part. The positive and negative contributions to the flux cancel.

Example with a closed surface

Consider a spherical surface placed around a bar magnet. Magnetic field lines cross the sphere in many places. Where the field points outward relative to the surface normal, the flux contribution is positive. Where the field points inward, the contribution is negative. Adding all contributions over the whole sphere gives zero.

Rendering scientific sketch…

Closed surface around a magnet

Differential form

The integral form describes the total flux through a closed surface. There is also a local, differential form:

$$
\nabla \cdot \vec{B} = 0
$$

This says that the divergence of the magnetic field is zero everywhere in ordinary electromagnetism. In simple terms, the magnetic field has no sources or sinks.

Differential form of Gauss's law for magnetism:
$$
\nabla \cdot \vec{B} = 0
$$
This means magnetic field lines do not start or stop at ordinary points in space.

Why this matters

Gauss's law for magnetism is one of Maxwell's equations. It expresses a deep symmetry of magnetic fields and helps distinguish magnetism from electricity. It also tells us what kinds of field patterns are possible. Any valid magnetic field must satisfy the condition that its net flux through a closed surface is zero.

This law is especially useful as a consistency check. If a proposed magnetic field would give nonzero net flux through a closed surface, then it cannot represent an ordinary physical magnetic field.

About magnetic monopoles

Physicists have wondered whether isolated magnetic charges, called magnetic monopoles, might exist. If they did, Gauss's law for magnetism would need to be modified, just as electric flux depends on electric charge. But in standard classical electromagnetism, and in all ordinary experiments at this level, no magnetic monopoles are included.

So for this course, the correct law is always

$$
\oint \vec{B}\cdot d\vec{A} = 0
$$

For standard electromagnetism in this course, magnetic monopoles are not present.
Therefore,
$$
\oint \vec{B}\cdot d\vec{A} = 0
$$
for every closed surface.

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