.. _theory_section:

Theory and Algorithms
=====================

This section provides a theoretical background on the computational methods used in the ``em-app`` package. It covers the core algorithm for magnetic field calculation, the role of the ``mtflib`` library in multipole expansions, and the fundamental principles from Maxwell's equations.

Core Algorithm: The Biot-Savart Law
------------------------------------

The primary method for calculating the magnetic field in this package is the Biot-Savart law. This law describes the magnetic field **B** generated by a constant electric current. For a one-dimensional wire carrying a current *I*, the magnetic field at a point **r** is given by the line integral:

.. math::

   \mathbf{B}(\mathbf{r}) = \frac{\mu_0 I}{4\pi} \int_C \frac{d\mathbf{l}' \times (\mathbf{r} - \mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|^3}

where:
- :math:`\mu_0` is the permeability of free space.
- :math:`I` is the current.
- :math:`d\mathbf{l}'` is a vector representing an infinitesimal segment of the wire.
- :math:`\mathbf{r}'` is the position vector of the segment :math:`d\mathbf{l}'`.
- :math:`\mathbf{r}` is the point where the magnetic field is being calculated.

In ``em-app``, complex current sources (like ``RingCoil`` or ``RectangularCoil``) are discretized into a finite number of straight-line segments. The total magnetic field is then calculated by summing the contributions from each segment, approximating the integral numerically.

Multipole Expansion using ``mtflib``
--------------------------------------

For points far from the source, calculating the magnetic field using the full Biot-Savart law can be computationally expensive. In these cases, a multipole expansion provides an accurate and efficient approximation. This expansion represents the complex magnetic field as a sum of simpler fields arising from multipoles (monopole, dipole, quadrupole, etc.).

The ``em-app`` package leverages the ``mtflib`` library to facilitate these calculations. ``mtflib`` allows for the representation of physical quantities, such as vector components and positions, as multivariate Taylor series objects. When operations are performed on these objects, the result is also a Taylor series.

This has two main advantages:

1.  **Automatic Differentiation**: The derivatives of the field (gradient, divergence, curl) can be computed automatically and with high precision.
2.  **Field Approximation**: The Taylor series itself is a power series representation of the field around a certain point. This is closely related to the multipole expansion, allowing for efficient field calculations at a distance.

By initializing calculations with ``mtf.initialize_mtf()``, users can compute not just the value of the magnetic field but also its spatial derivatives, which are essential for understanding the field's local structure.

Maxwell's Equations and Field Properties
----------------------------------------

The behavior of electric and magnetic fields is governed by Maxwell's equations. In their differential form, they are:

1.  **Gauss's Law for Electricity**: :math:`\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}`
2.  **Gauss's Law for Magnetism**: :math:`\nabla \cdot \mathbf{B} = 0`
3.  **Faraday's Law of Induction**: :math:`\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}`
4.  **Ampère-Maxwell Law**: :math:`\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)`

From these fundamental equations, we can derive key properties of the magnetic field in magnetostatics (the study of steady currents).

### Divergence-Free Magnetic Field

In magnetostatics, we assume that the fields are not changing in time (:math:`\frac{\partial}{\partial t} = 0`). Gauss's Law for Magnetism, :math:`\nabla \cdot \mathbf{B} = 0`, holds universally. This means that the magnetic field is **divergence-free**.

Physically, this implies that there are no magnetic monopoles (isolated north or south poles). Magnetic field lines always form closed loops.

### Curl of the Magnetic Field

In magnetostatics, the Ampère-Maxwell Law simplifies to:

.. math::

   \nabla \times \mathbf{B} = \mu_0 \mathbf{J}

This equation states that the curl of the magnetic field is proportional to the current density **J**. The curl measures the "rotation" of the field at a point.

In a region of space where there are **no current sources** (:math:`\mathbf{J} = 0`), the equation further simplifies to:

.. math::

   \nabla \times \mathbf{B} = 0

This means that in a source-free region, the magnetic field is **curl-free**. A curl-free vector field can be expressed as the gradient of a scalar potential, which simplifies many calculations. The ``em-app`` package allows for the direct computation of the curl and divergence of the field, enabling users to verify these fundamental properties.