Magnetic flux density

The magnetic flux density $\mathbf{B}$ is the vector field that produces forces on moving charges via the Lorentz force:

$\mathbf{F}=q\mathbf{u}\times \mathbf{B}\text{(N)},$

with units of tesla (T = Wb/m² = N/(A·m) = kg/(A·s²)). $\mathbf{B}$ is what charges in motion actually “feel” — it includes contributions from both free currents and bound magnetisation currents in magnetic materials.

The constitutive relation in a linear, isotropic, homogeneous medium relates $\mathbf{B}$ to the magnetic field intensity $\mathbf{H}$:

$\mathbf{B}=\mu \mathbf{H},$

where $\mu ={\mu }_r{\mu }_0$ is the Permeability. For free space ${\mu }_0=4\pi \times 10^{-7}$ H/m. For non-magnetic materials ${\mu }_r\approx 1$, so $\mathbf{B}\approx {\mu }_0\mathbf{H}$ — the two fields are interchangeable up to a constant.

Why two magnetic fields?

The same logic that motivates the Electric flux density/Electric field split. $\mathbf{H}$ is the “free-source” field: $\nabla \times \mathbf{H}={\mathbf{J}}_{\text{free}}$ contains only free currents. $\mathbf{B}$ is the “what charges feel” field: it includes bound magnetisation currents in magnetic materials.

In a magnetised material, atomic currents (electron orbital and spin moments) produce an additional magnetic contribution. Those bound currents are absorbed into $\mu$ via $\mathbf{B}=\mu \mathbf{H}$, leaving $\mathbf{H}$ depending only on free currents.

For non-magnetic problems (most of EM in vacuum or air), the distinction collapses and engineers use the two fields almost interchangeably.

Gauss’s law for magnetism

Gauss’s law for magnetism states

$\nabla \cdot \mathbf{B}=0⟺{\oint }_S\mathbf{B}\cdot d\mathbf{s}=0$

for any closed surface $S$. Physical content: there are no magnetic monopoles. Every $\mathbf{B}$ field line is a closed loop — flux into any closed surface equals flux out.

Compare with Electric flux density: $\nabla \cdot \mathbf{D}={\rho }_v$ — electric flux begins on positive charges and ends on negative charges. Magnetic flux has no such sources or sinks.

Magnetic vector potential

Because $\nabla \cdot \mathbf{B}=0$, $\mathbf{B}$ is solenoidal (divergence-free), and a vector identity guarantees the existence of a Vector magnetic potential $\mathbf{A}$ such that

$\mathbf{B}=\nabla \times \mathbf{A}.$

This is analogous to writing the electric field as $\mathbf{E}=-\nabla V$ in electrostatics — except the potential is now a vector field, not a scalar, because $\mathbf{B}$ is divergence-free rather than curl-free.

Vector potential plays a central role in:

• Computing $\mathbf{B}$ from current distributions via $\mathbf{A}=(\mu /4\pi )\int (\mathbf{J}/R)d{v}^{\prime }$.
• Quantum mechanics (the Aharonov-Bohm effect).
• Field theory (gauge invariance).

Magnetic flux

The total magnetic flux $\Phi$ through a surface $S$ is

$\Phi ={\iint }_S\mathbf{B}\cdot d\mathbf{s}\text{(Wb)},$

with units of weber (Wb = T·m²). This is what appears in Faraday’s law:

$\text{emf}=-\frac{d\Phi }{dt}.$

A changing magnetic flux through a circuit induces an electromotive force, the operating principle of generators, transformers, and inductors. So the time derivative of $\mathbf{B}$ (integrated over a surface) is what couples back to $\mathbf{E}$ — the link between the static magnetic and dynamic electric pictures.

Magnitudes you should recognise

| Source | Approximate $|\mathbf{B}|$ | |--------|---------------------------| | Earth’s surface field | $\sim 50\mu$T | | Refrigerator magnet | $\sim 5$mT | | Strong NdFeB permanent magnet | $\sim 1$T | | Medical MRI scanner | $1.5$–$7$T | | Strongest sustained lab field | $\sim 45$T (continuous), $\sim 1200$T (pulsed, destructive) | | Surface of a neutron star | $\sim 10^8$T |

A 1-T field is “very strong” by everyday standards. The MRI cylinder you walk into is one of the largest sustained magnetic fields humans encounter routinely.

In Maxwell’s equations

The two magnetic equations:

• $\nabla \cdot \mathbf{B}=0$ — no monopoles. (Solenoidal $\mathbf{B}$.)
• $\nabla \times \mathbf{H}=\mathbf{J}+{\partial }_t\mathbf{D}$ — Ampère with Displacement current. (The $\mathbf{B}$ form: $\nabla \times \mathbf{B}={\mu }_0(\mathbf{J}+{ϵ}_0{\partial }_t\mathbf{E})$ in free space.)

Together with Gauss’s law for $\mathbf{D}$ and Faraday’s law (which involves ${\partial }_t\mathbf{B}$), these are the four classical Maxwell equations.

Versus electric flux density

A close parallel:

Concept	Electric	Magnetic
”Free-source” field	Electric flux density $\mathbf{D}$	Magnetic field intensity $\mathbf{H}$
“What charges feel” field	Electric field $\mathbf{E}$	Magnetic flux density $\mathbf{B}$
Constitutive relation	$\mathbf{D}=ϵ\mathbf{E}$	$\mathbf{B}=\mu \mathbf{H}$
Material parameter	Permittivity $ϵ$	Permeability $\mu$
Gauss’s law	$\nabla \cdot \mathbf{D}={\rho }_v$	$\nabla \cdot \mathbf{B}=0$
Sources	Free electric charges	None (no magnetic monopoles)
Force law	$\mathbf{F}=q\mathbf{E}$	$\mathbf{F}=q\mathbf{u}\times \mathbf{B}$
Potential	scalar $V$, $\mathbf{E}=-\nabla V$	vector $\mathbf{A}$, $\mathbf{B}=\nabla \times \mathbf{A}$

The asymmetry — Gauss’s law on the right but no monopoles on the left — is one of the deep open questions in physics. Dirac showed that the existence of even one magnetic monopole would explain charge quantisation; experimentally, none have been found.

In context

Magnetic flux density is one half of the magnetic-field pair; the other is magnetic field intensity $\mathbf{H}$. Together they describe magnetostatic and dynamic fields throughout EM. Related concepts: Magnetic flux (surface integral of $\mathbf{B}$), Vector magnetic potential $\mathbf{A}$, Faraday’s law (relating ${\partial }_t\mathbf{B}$ to induced electric fields).