Gauss's Law for Magnetism

Equivalent forms

\nabla \cdot \vec{B} = 0

\Phi_{B,\text{closed}} = 0

The simplest of Maxwell's equations — a single zero that says magnetic charges do not exist (or at least have never been found).

Symbol	Name	SI unit	Dimension	Range
$Φ_B$	Magnetic flux (closed surface)output Net magnetic flux through any closed surface	Wb	M·L²·T⁻²·I⁻¹	0 – 0
$B$	Magnetic field Local magnetic field vector	T	M·T⁻²·I⁻¹	0 – 10

Unit systems

Symbol	SI	CGS	Imperial
$Φ_B$	Wb	Mx	Wb
$B$	T	G	T

Where it holds

Universally valid in classical and quantum electrodynamics. Would be modified if magnetic monopoles are discovered. One of the four Maxwell equations — holds for all electromagnetic phenomena.

Dimensional analysis

$\oint [B]\cdot [dA] \to T\cdot m^{2} = Wb = 0 \checkmark$

Discovery

Carl Friedrich Gauss / James Clerk Maxwell · 1835

Gauss recognized the divergence-free nature of magnetic fields from experimental observations. Maxwell elevated it to one of his four fundamental equations, encoding the non-existence of magnetic monopoles.

Try this

Why can't you isolate a magnetic north pole by cutting a bar magnet in half?

Explain why the total magnetic flux through any closed surface is always zero, and what this implies for magnetic field lines.

Research status: stable

Real-world applications

Magnetic circuit analysis — ensures flux conservation in transformer cores
MRI field homogeneity design — B lines must close, constraining shimming strategies
Geomagnetic modeling — Earth's field must satisfy $\nabla \cdot B = 0$ at every point
Motivating ongoing experimental searches for magnetic monopoles

Common misconceptions

This law does NOT $say B = 0$ anywhere — it says the divergence of B is zero everywhere
Cutting a magnet in half does not create a monopole — you get two smaller dipoles
The law applies to the net flux through a closed surface, not through an open surface

Experimental verification

No magnetic monopole has ever been detected despite extensive searches (cosmic ray experiments, accelerator searches, geological samples). Upper bounds on monopole flux set by MACRO, IceCube, and MoEDAL experiments at CERN.

Derivation

From the Biot–Savart law,

B = (\mu _{0}/4\pi )\int (J \times r

̂/

r^{2})dV

.

Taking the divergence:

\nabla \cdot B = 0

because

\nabla \cdot (J \times r

̂/

r^{2}) = 0

identically (a vector calculus identity).

Physically: every magnetic field line is a closed loop, so flux entering any volume must equal flux exiting.

Limiting cases

Any closed surface, any field⟶

\oint B\cdot dA = 0

No exceptions in classical electromagnetism — this is an absolute law.

If magnetic monopoles existed⟶

\oint B\cdot dA = \mu _{0}\cdot q_m

Hypothetical modification — would make Maxwell's equations more symmetric but none have been found.

What if…

What if you cut a bar magnet in half?

You get two smaller magnets, each with N and S poles. No monopole is created. $\oint B\cdot dA = 0$ still holds for any surface around either piece.

What if magnetic monopoles existed?

The law would become $\oint B\cdot dA = \mu _{0}q_m$ (like Gauss's law for E). Maxwell's equations would gain perfect symmetry between E and B. String theory predicts they may exist — searches continue at CERN.

What if you measure nonzero flux through a 'closed' surface?

Your surface has a gap or your measurement has errors. In classical EM this is impossible — it would be as revolutionary as discovering a new fundamental particle.

1

Flux through a closed surface around a bar magnet

Given ·

description:: A bar magnet inside a closed rectangular box

Find · Net magnetic flux through the box

Steps

Gauss's law for magnetism: $\oint B\cdot dA = 0$ for any closed surface
Field lines exit the north pole and enter the south pole
Every field line that exits through one face of the box must re-enter through another
Net outward flux $= net$ inward flux → total net flux $= 0$
This holds regardless of the magnet's size, strength, or position inside the box

Result ·

\Phi _B = 0

(always, for any closed surface)

2

Demonstrating ∇·B = 0 for a solenoid field

Given ·

B inside:: 0.01
solenoid radius:: 0.05

Find · Verify divergence-free condition

Steps

Inside the solenoid: B is uniform and axial $\to \partial B_z/\partial z = 0$
No radial or azimuthal components inside $\to \partial B_r/\partial r = 0$ , $\partial B_\theta /\partial \theta = 0$
$\nabla \cdot B = 0$ inside $\checkmark$
Outside: field lines curve from north to south end, closing the loops
At every point outside, the divergence also vanishes — flux in equals flux out for any volume element

Result ·

\nabla \cdot B = 0

everywhere — field lines that run straight inside the solenoid curve around outside to close on themselves

Gauss's Law→Ampère's Law (with Maxwell's correction)→Faraday's Law of Induction→