Equation Models

This project uses analytical field expressions and geometric reductions rather than a generic PDE solve. The kernels are organized by source family, but they share a common field convention and local-coordinate reduction strategy.

Field conventions

The code distinguishes four quantities:

  • magnetic flux density: $\mathbf{B}$

  • magnetic field strength: $\mathbf{H}$

  • polarization: $\mathbf{J}$

  • magnetization: $\mathbf{M}$

For linear vacuum conversions used throughout the library:

$$ \mathbf{B} = \mu_0 \mathbf{H}, \qquad \mathbf{J} = \mu_0 \mathbf{M}. $$

For permanent magnets, the implementation usually computes a field representation in terms of polarization and then converts to the requested output quantity.

Common reduction pattern

Most kernels follow the same three-step structure:

  1. Transform observers into a local frame attached to the source.

  2. Evaluate a closed-form field model in that frame.

  3. Rotate the result back to global coordinates.

If the source has a motion path, the reduction is repeated path-wise with Magpylib-compatible broadcasting and squeeze semantics.

Dipole model

For dipole moment $\mathbf{m}$ and observation vector $\mathbf{r}$:

$$ \mathbf{H}(\mathbf{r}) = \frac{1}{4\pi} \left( \frac{3(\mathbf{m}\cdot\mathbf{r})\mathbf{r}}{\lVert \mathbf{r} \rVert^5}

\frac{\mathbf{m}}{\lVert \mathbf{r} \rVert^3} \right), $$

and then

$$ \mathbf{B}(\mathbf{r}) = \mu_0 \mathbf{H}(\mathbf{r}). $$

This is the far-field reference used explicitly for misc.Dipole and implicitly as a limiting comparison for other source families.

Sphere with uniform polarization

A uniformly polarized sphere has a particularly simple decomposition:

  • inside the sphere, the magnetic field is uniform,

  • outside the sphere, the field is equivalent to that of a dipole with moment proportional to the sphere volume.

With polarization $\mathbf{J}$:

$$ \mathbf{B}_{\text{inside}} = \frac{2}{3}\mathbf{J}, $$

while outside the sphere the implementation uses the dipole-equivalent form.

Current line and polyline models

For line currents the code uses Biot-Savart closed forms on each segment and sums them.

For a current element,

$$ \mathrm{d}\mathbf{B} = \frac{\mu_0 I}{4\pi} \frac{\mathrm{d}\mathbf{l} \times \mathbf{r}}{\lVert \mathbf{r} \rVert^3}. $$

The polyline implementation reduces the source to a set of finite segments and accumulates the analytical segment fields. This gives a path that is exact for the piecewise-linear geometry and differentiable away from the segment singular set.

Circular loop model

The circle kernel uses the standard cylindrical-coordinate analytical reduction with complete elliptic integrals. The implementation relies on a robust elliptic helper in core/elliptic.py and handles the axis separately to avoid unstable generic formulas.

The important implementation point is not the textbook formula itself, but the combination of:

  • cylindrical reduction,

  • axis masks,

  • x64-friendly evaluation,

  • JAX-compatible branching.

Cuboid, triangle, tetrahedron, and mesh models

Permanent-magnet kernels are built from surface-charge or solid-angle formulations.

Triangle

The triangle kernel is the basic surface element. For a uniformly polarized planar triangle, the field can be written in terms of edge contributions and a solid-angle term. That representation is reused directly or indirectly by several higher-level sources.

Tetrahedron

A tetrahedron is reduced to a sum over oriented triangle faces. The implementation precomputes shared geometry terms per face and accumulates them in JAX, which improves both memory behavior and compilation reuse.

Triangular mesh

A TriangularMesh is reduced to its oriented faces. The inside/outside behavior, open-mesh handling, and face orientation semantics are treated as compatibility-sensitive behavior and are validated against upstream Magpylib.

TriangleSheet and TriangleStrip

Current-sheet kernels map each triangle into a canonical local frame with vertices of the form

$$ (0,0,0), \quad (u_1,0,0), \quad (u_2, v_2, 0), $$

then apply the analytical sheet-field expressions triangle by triangle. Edge, in-plane, and off-plane limits are handled separately because the physically correct limit is piecewise.

Cylinder and CylinderSegment models

The cylinder family combines analytical radial/axial reductions with explicit boundary handling.

Cylinder

The full cylinder kernel is treated as a rotationally symmetric magnet and uses cylindrical reduction with elliptic-integral building blocks.

CylinderSegment

CylinderSegment is more difficult because symmetry is reduced: the source has radial, axial, and azimuthal boundaries. The implementation therefore uses a face-based decomposition with precomputed geometry and specialized JIT entrypoints for hotspot profiling.

Derivation strategy in code

The codebase does not try to reproduce textbook derivations line by line inside docstrings. Instead, it encodes derivation structure as reusable geometry reductions:

  • local-frame transforms,

  • oriented face decompositions,

  • edge and solid-angle terms,

  • stable branch masks for singular neighborhoods.

That structure is easier to validate and profile than directly transcribing symbolic expressions into monolithic formulas.

Where to read the implementation