Feel++

Maxwell’s equations

We present here the Maxwell equations

Differential form of Maxwell equations

The differential form of Maxwell’s equations reads:

\$\begin{aligned} \nabla \times \mathbf{H}(\mathbf{x},t) &= \mathbf{J}(\mathbf{x},t) + \frac{\partial \mathbf{D}(\mathbf{x},t)}{\partial t}\\ \nabla \times \mathbf{E}(\mathbf{x},t) & = - \frac{\partial \mathbf{B}(\mathbf{x},t)}{\partial t}\\ \nabla \cdot \mathbf{B}(\mathbf{x},t) & = 0\\ \nabla \cdot \mathbf{D}(\mathbf{x},t) & = \rho(\mathbf{x},t)\\ \mathbf{B}(\mathbf{x},t) &= \mu_0[\mathbf{H}(\mathbf{x}, t) + \mathbf{M}(\mathbf{x}, t)],\\ \mathbf{J}(\mathbf{x},t) &= \sigma[\mathbf{E}(\mathbf{x}, t) + \mathbf{E}_i(\mathbf{x}, t)] = \sigma \mathbf{E}(\mathbf{x}, t) + \mathbf{J}_i(\mathbf{x}, t)\\ \mathbf{D}(\mathbf{x},t) & = \epsilon_0 \mathbf{E}(\mathbf{x}, t) + \mathbf{P}(\mathbf{x}, t) \end{aligned}\$

Variables, symbols and units

The following table provides the names and units (in SI) of the symbols and variables above.

Table 1. Name and units of symbols and variables of the Maxwell’s equations
Notation Quantity Unit SI

\$\mathbf{H}(\mathbf{x},t)\$

magnetic field intensity

\$A\cdot m^{-1}\$

\$A\cdot m^{-1}\$

\$\mathbf{E}(\mathbf{x},t)\$

electric field intensity

\$V\cdot m^{-1}\$

\$kg\cdot m \cdot s^{-3}\cdot A^{-1}\$

\$\mathbf{B}(\mathbf{x},t)\$

magnetic flux density

\$T\$

\$kg\cdot s^{-2}\cdot A^{-1}\$

\$\mathbf{D}(\mathbf{x},t)\$

electric flux density

\$C\cdot m^{-2}\$

\$A\cdot s\cdot m^{-2}\$

\$\mathbf{A}(\mathbf{x},t)\$

magnetic potential vector

\$V\cdot s\cdot m^{-1}\$

\$kg\cdot m \cdot s^{-2}\cdot A^{-1}\$

\$\mathbf{J}(\mathbf{x},t)\$

electric current density

\$A\cdot m^{-2}\$

\$A\cdot m^{-2}\$

\$\rho(\mathbf{x},t)\$

electric charge density

\$C\cdot m^{-3}\$

\$A\cdot s\cdot m^{-3}\$

\$\mathbf{M}(\mathbf{x},t)\$

magnetisation

\$A\cdot m^{-1}\$

\$A\cdot m^{-1}\$

\$\mathbf{E}_i(\mathbf{x},t)\$

impressed electric field

\$V\cdot m^{-1}\$

\$kg\cdot m \cdot s^{-3}\cdot A^{-1}\$

\$\mathbf{J}_i(\mathbf{x},t)\$

impressed electric current

\$A\cdot m^{-2}\$

\$A\cdot m^{-2}\$

\$\mathbf{P}(\mathbf{x},t)\$

polarization

\$C\cdot m^{-2}\$

\$A\cdot s\cdot m^{-2}\$

\$\mu_0(\mathbf{x},t)\$

permeability of vacuum

\$H\cdot m^{-1}\$

\$kg\cdot m\cdot s^{-2}\cdot A^{-2}\$

\$\sigma(\mathbf{x},t)\$

conductivity

\$S\cdot m^{-1}\$

\$kg^{-1}\cdot m^{-3}\cdot s^3\cdot A^2\$

\$\epsilon_0(\mathbf{x},t)\$

permittivity of vacuum

\$F\cdot m^{-1}\$

\$kg^{-1}\cdot m^{-3}\cdot s^4\cdot A^2\$

The field quantities are depending on space \$\mathbf{x}\$ and on time \$t\$, we will omit later both in the notation and use e.g. \$\mathbf{H}, \mathbf{E}, \ldots\$

The sources of the electromagnetic fields are the electric current density \$\mathbf{J}\$ and the electric charge density \$\rho\$.

Constitutive relations

The last three equations in Maxwell’s equations above collect the constitutive relations, which – depending on the properties of the examined material – describe the relationship between field quantities.

In the simplest case these relations are linear, i.e.

\$\mathbf{M} = \chi\mathbf{H},\quad \mathbf{E}_i=0,\quad \mathbf{P}=\epsilon_0\chi_d\mathbf{E}\$

where \$\chi\$ and \chi_d are the magnetic and the dielectric susceptibility respectively and

\$\mathbf{B} = \mu \mathbf{H},\quad \mathbf{J} = \sigma \mathbf{E}, \quad \mathbf{D} = \epsilon_0 \mathbf{E}\$

where

\$\mu = \mu_0(1 + \chi) = \mu_0\mu_r,\quad \epsilon = \epsilon_0(1 + \chi_d) = \epsilon_0\epsilon_r.\$

Here \$\mu_r = 1 + \chi\$ is the relative permeability, \$\epsilon_r = 1 + \chi_d\$ is the relative permittivity of the material and the conductivity \$\sigma\$ is constant. The equation \$\mathbf{J} = \sigma \mathbf{E}\$ is the differential form of Ohm’s law.

The constitutive relations are nonlinear in general, that is the permeability, conductivity and permittivity depend on the associated field quantities, i.e.

\$\mu = \mu(\mathbf{H},\mathbf{B}),\quad \sigma = \sigma(\mathbf{E}, \mathbf{J}),\quad \epsilon = \epsilon(\mathbf{E}, \mathbf{D})\$

or equivalently

\$\mathbf{B} = \mathcal{B}(\mathbf{H}),\quad \mathbf{J} = \mathcal{J}(\mathbf{E}),\quad \mathbf{D} = \mathcal{D}(\mathbf{E})\$

where \mathcal{B}(·), \mathcal{J}(·) \text{ and } \mathcal{D}(·) are operators. We would actually need also the inverse operator, e.g.

\$\mathbf{H} = \mathcal{B}^{-1}(\mathbf{B})\$

Classification of Maxwell’s Equations

We now turn to the classification of the Maxwell’s equations.

Steady case

The simplest case corresponds to the steady case: the time variation of the field quantities can be neglected, i.e. \partial/\partial t = 0. The corresponding fields are called static field. In this case the magnetic, the electric and the current fields can be treated independently because there are no interactions between them.

Time varying case

When \$\partial/\partial t = 0\$, the magnetic and electric fields are coupled, we are then in the presence of eddy current fields and wave propagation of electrodynamics.

Static magnetic field

The time independent current density \$\mathbf{J} = \mathbf{J}(\mathbf{x})\$ generates the time independent magnetic field intensity \$\mathbf{H} = \mathbf{H}(\mathbf{x})\$ and the time independent magnetic flux density \$\mathbf{B} = \mathbf{B}(\mathbf{x})\$.

the magnetostatic Maxwell’s equations read

\$\begin{aligned} \nabla \times \mathbf{H}(\mathbf{x}) &= \mathbf{J}(\mathbf{x}) \\ \nabla \cdot \mathbf{B}(\mathbf{x}) & = 0\\ \mathbf{B}(\mathbf{x}) &= \begin{cases} \mu_0 \mathbf{H}\,& \text{ in air}\\ \mu_0\mu_r \mathbf{H}\,& \text{ in magnetically linear media}\\ \mu_0[\mathbf{H} + \mathbf{M}]\,& \text{ in magnetically nonlinear media} \end{cases} \end{aligned}\$

In a nonlinear medium, the magnetization vector \$\mathbf{M} = \mathbf{M}(\mathbf{x})\$ is depending on the magnetic field intensity vector, i.e. \$\mathbf{M} = \mathcal{H}(\mathbf{H})\$.

The operator \$\mathcal{H}\$ can be described by so-called hysteresis models denoted by \$\mathbf{B} = \mathcal{B}(\mathbf{H})\$.

This constitutive relation has an inverse form which read

\$\mathbf{H} = \begin{cases} \nu_0 \mathbf{B}\,& \text{ in air}\\ \nu_0\nu_r \mathbf{B}\,& \text{ in magnetically linear media}\\ \mathcal{B}^{-1}(\mathbf{B})\,& \text{ in magnetically nonlinear media} \end{cases}\$

where \$\nu_0 = 1/\mu_0,\, \nu_r = 1/\mu_r\$ are the reluctivity of vacuum and the relative reluctivity.

In magnetically nonlinear media, it can be represented by an inverse hysteresis operator, \$\mathbf{H} = \mathcal{B}^{-1}(\mathbf{B}).\$

The source current distribution is solenoidal, which reads \$\nabla \cdot \mathbf{J} = 0\$ (take the divergence of the first Maxwell’s equation). This means that all current lines either close upon themselves, or start and terminate at infinity.
This case corresponds to magnetic fields generated by (i) coils carrying currents or (ii) the static behavior of electrical machines. When \$\mathbf{J}=0\$, then a boundary value problem to simulate e.g. the field of magnetic poles.

Magnetostatic problem formulation

Denote \$\Omega_0\$ the non-magnetic (e.g. air) part of \$\Omega\$ (hence 0) and \$\Omega_m\$ the magnetic part.

In the case of static magnetic field, the Maxwell’s equations read

\$\begin{aligned} \nabla \times \mathbf{H}(\mathbf{x}) &= \mathbf{J}(\mathbf{x}) \text{ in } \Omega_0 \cup \Omega_m\\ \nabla \cdot \mathbf{B}(\mathbf{x}) & = 0 \text{ in } \Omega_0 \cup \Omega_m\\ \mathbf{B}(\mathbf{x}) &= \begin{cases} \mu_0 \mathbf{H}\,& \text{ in air}\\ \mu_0\mu_r \mathbf{H}\,& \text{ in magnetically linear media}\\ \mathcal{B}(\mathbf{H}) = \mu_{\mathrm{o}} \mathbf{H}+\mathbf{R}\,& \text{ in magnetically nonlinear media} \end{cases} \end{aligned}\$

The constitutive relation has an inverse form

\$\begin{aligned} \mathbf{H}(\mathbf{x}) &= \begin{cases} \nu_0 \mathbf{B}\,& \text{ in air}\\ \nu_0\nu_r \mathbf{B}\,& \text{ in magnetically linear media}\\ \mathcal{B}^{-1}(\mathbf{B}) = \nu_{\mathrm{o}} \mathbf{B}+\mathbf{I}\,& \text{ in magnetically nonlinear media} \end{cases} \end{aligned}\$

where \$\mu_{\mathrm{o}}\$ and \$\nu_{\mathrm{o}}\$ are the optimal permeability and reluctivity respectively obtained using the polarisation method described below.

Only the tangential components of \$\mathbf{H}\$ is continuous across the interface \$\Gamma_{0m}\$ between \$\Omega_0\$ and \$\Omega_m\$. As to \$\mathbf{B}\$, it is its normal component which is continuous across \$\Gamma_{0m}\$.

Vector Potential formulation for magnetostatic

The magnetic vector potential \$\mathbf{A}\$ is defined by

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

which satisfies \$\nabla \cdot \mathbf{B} = 0\$ exactly, because of the identity \$\nabla \cdot \nabla \times \mathbf{v} = 0\$ for any vector function \$\mathbf{v}\$.

To ensure the uniqueness of the magnetic vector potential, its divergence can be selected according to Coulomb gauge,
\$\nabla \cdot \mathbf{A} = 0\$

This is useful, because the vector potential \$\mathbf{A}' = \mathbf{A} + \nabla \phi\$ also satisfies the equations above, because of the identity \$\nabla \times \nabla \phi = \mathbf{0}\$ where \$\phi\$ is a scalar field. This is the reason why the magnetic vector potential is not unique.

Substituting the definition of \$\mathbf{A}\$ into the first Maxwell’s equation and using the linearized constitutive relation, we get

\$\nabla \times (\nu_{\mathrm{o}} \nabla \times \mathbf{A} ) = \mathbf{J} - \nabla \times \mathbf{I}\quad \text{ in } \Omega\$

where \$\mathbf{J}\$ is the source current density.

In case where the media is linear, the term \$-\nabla\times\mathbf{I}\$ disappears.

The strategy to solve this equation is discussed in the Strategy Chapter.

Benchmark

We benchmark here our implementation.

We set - for convenience - \$\mu_{\mathrm{o}}\$ to one in that convergence test.

Given a sinusoïdal solution, we compute - with no regularization terms (we are not interested in the potential vector but its curl) - the appropriate right hand side and use the exact solution a boundary condition.

\$\begin{aligned} \mathbf{J}&= \begin{pmatrix} 3 \pi^3 \cos(\pi x) \sin(\pi y)\sin(\pi z) \\ -6\pi^3 \sin(\pi x) \cos(\pi y) \sin(\pi z) \\ 3\pi^3 \sin(\pi x) \sin(\pi y) \cos(\pi z) \end{pmatrix} \\ \mathbf{A}_{exact}&=\begin{pmatrix} \pi \cos(\pi x) \sin(\pi y) \sin(\pi z)\\ -2\pi \sin(\pi x) \cos(\pi y) \sin(\pi z) \\ \pi \sin(\pi x) \sin(\pi y) \cos(\pi z)\end{pmatrix} \\ \mathbf{c}&=\begin{pmatrix}3 \pi^2 \cos(\pi z) \cos(\pi y)\sin(\pi x)\\0 \\-(3\pi^2) \sin(\pi z) \cos(\pi y)\cos(\pi x )\end{pmatrix} \end{aligned}\$

Regularized point system

\$\begin{aligned} \nabla \times \left(\frac{1}{\mu_{\mathrm{o}}} \nabla \times \mathbf{A} \right) + \epsilon \mathbf{A} &= \mathbf{J} \quad \text{ in } \Omega \\ \left.\mathbf{A}\right|_{\partial \Omega} &= \mathbf{A}_{exact} \\ \end{aligned}\$

Saddle point system

\$\begin{aligned} \nabla \times \left(\frac{1}{\mu_{\mathrm{o}}} \nabla \times \mathbf{A} \right) + \nabla p &= \mathbf{J} \quad \text{ in } \Omega \\ \nabla \cdot \mathbf{A} &= 0 \quad\text{ in } \Omega \\ \left.\mathbf{A}\right|_{\partial \Omega} &= \mathbf{A}_{exact} \\ \left.p\right|_{\partial \Omega} &= 0 \end{aligned}\$

The boundary condition can apply with penalization or elimination. We compare both results: Saddle Point system convergence