Solenoidal magnetic fields and no magnetic monopoles

The solenoidal law for magnetism is:

$$\nabla \cdot \mathbf{B}=0.$$

It says magnetic fields have no sources or sinks inside space. Field lines do not begin or end inside the volume; they either form closed loops or pass through the boundary.

This is also called Gauss’s law for magnetism.

Electric fields are different

The electric version is:

$$\nabla \cdot \mathbf{E}=\frac{\rho }{{ϵ}_0}.$$

Electric charge can act as a source or sink:

• positive charge is a source of electric field;
• negative charge is a sink of electric field;
• electric field lines can begin and end on charges.

If magnetism had isolated magnetic charges, we might write:

$$\nabla \cdot \mathbf{B}={\mu }_0{\rho }_m,$$

where ${\rho }_m$ would be a magnetic charge density.

But isolated magnetic north or south charges have not been observed in standard electromagnetism, so:

$${\rho }_m=0,$$

and therefore:

$$\nabla \cdot \mathbf{B}=0.$$

Intuition

For electric fields, a tap-and-drain picture works:

• positive charge: tap/source;
• negative charge: drain/sink;
• field lines can begin and end.

For magnetic fields, there are no taps or drains. The field is more like closed circulation.

Cut a bar magnet in half and it does not produce one isolated north magnet and one isolated south magnet. It produces two smaller dipoles:

$$\text{N–S}⟶\text{N–S}+\text{N–S}.$$

That is the everyday version of “no magnetic monopoles”.

Why this matters for NF2

NF2 predicts a three-dimensional magnetic field $\mathbf{B}(x,y,z)$. A physically credible field should satisfy:

$$\nabla \cdot \mathbf{B}\approx 0.$$

If the divergence is large, the numerical field contains artificial magnetic sources or sinks. Those are not solar physics; they are artefacts of the model, data handling, or numerical method.

Large divergence can damage:

• field-line topology;
• magnetic connectivity;
• current-density estimates;
• magnetic-energy estimates;
• comparisons with potential-field or NLFFF baselines.

So NF2 and other PINN-style models include a divergence residual:

$$R_{\mathrm{div}}=\nabla \cdot \mathbf{B},$$

and often minimise a loss such as:

$${\mathcal{L}}_{\mathrm{div}}=\frac{1}{N}\sum _{i=1}^N{\left|\nabla \cdot \mathbf{B}({\mathbf{x}}_i)\right|}^2.$$

This term is the model being penalised for inventing magnetic monopoles.

Related

• Residuals in NF2 and PINNs
• NF2 solar extrapolation MOC
• HMI SHARP CEA vector components
• Magnetic topology and field-line tracing