General Material¶

Introduction¶

A material in the time-harmonic Maxwell model is characterized by its magnetic permeability $\mu$, electric permittivity $\varepsilon$, and electric conductivity $\sigma$. A general material can be introduced using specific material property methods as follows:

my_material = TimeHarmonicMaxwellGeneralMaterial(
   name = "My Material",
   marker = my_material_marker,
   permeability_method = my_permeability_method,
   permittivity_method = my_permittivity_method,
   conductivity_method = my_conductivity_method,
)

Material Properties¶

Magnetic Permeability¶

The magnetic permeability $\mu$ characterizes the response of a medium to an applied magnetic field and serves as the proportionality factor relating the magnetic field vectors $\tilde{\mathbf{B}}$ and $\tilde{\mathbf{H}}$: $$ \begin{align} \tilde{\mathbf{B}} = \mu \tilde{\mathbf{H}}. \end{align} $$

It commonly expressed as $$ \begin{align} \mu = \mu_0 \mu_r, \end{align} $$

where $\mu_0$ denotes the magnetic permeability of free space and $\mu_r$ is the relative magnetic permeability of the medium.

In the most general case, the magnetic permeability $\mu$ may depend on the radiation frequency, spatial position, or propagation direction within the medium. However, for most materials and practical applications described by the time-harmonic Maxwell model, $\mu$ is approximately equal to the vacuum permeability $\mu_0$, corresponding to a relative magnetic permeability $\mu_r=1$.

The magnetic permeability can be defined by the following methods:

Magnetic Permeability Methods
Name	Description
Linear Isotropic	`ComplexMagneticPermeabilityMethodLinearIsotropic` method specifies a constant relative magnetic permeability $\mu_r$.

Electric Permittivity¶

The electric permittivity $\varepsilon$ quantifies the response of a medium to an applied electric field and determines the relation between the electric displacement field $\tilde{\mathbf{D}}$ and the electric field $\tilde{\mathbf{E}}$: $$ \begin{align} \tilde{\mathbf{D}} = \varepsilon \tilde{\mathbf{E}}. \end{align} $$

It is conventionally written as $$ \begin{align} \varepsilon = \varepsilon_0 \varepsilon_r, \end{align} $$

where $\varepsilon_0$ is the permittivity of free space and $\varepsilon_r$ denotes the relative permittivity of the material.

In general, the permittivity $\varepsilon$ may vary with frequency, position, or direction within the medium, reflecting dispersive, inhomogeneous, or anisotropic behavior.

For many dielectric materials, the electric permittivity exceeds the vacuum value only moderately, with relative permittivity typically in the range $\varepsilon_r \approx 2 - 10$. Materials with stronger polarization response, such as polar dielectrics, may exhibit larger values, while metals can be described by an effectively very large and frequency-dependent permittivity, which dominates their electromagnetic behavior.

The electric permittivity can be defined by the following methods:

Electric Permittivity Methods
Name	Description
Linear Isotropic	`ComplexElectricPermittivityMethodLinearIsotropic` method specifies a constant relative electric permittivity $\varepsilon_r$.

Electric Conductivity¶

The electric conductivity $\sigma$ characterizes the ability of a medium to conduct electric current in response to an applied electric field. It relates the electric field $\tilde{\mathbf{E}}$ to the induced current density $\tilde{\mathbf{J}}$ through Ohm's law: $$ \begin{align} \tilde{\mathbf{J}} = \sigma \tilde{\mathbf{E}}. \end{align} $$

In general, the conductivity $\sigma$ may depend on frequency, spatial position, or direction within the material, reflecting dispersive, inhomogeneous, or anisotropic transport properties. In the time-harmonic electromagnetic models, $\sigma$ accounts for dissipative losses and contributes to the attenuation of electromagnetic waves in conductive media.

For many dielectric materials, the conductivity is negligibly small and can often be approximated as zero, whereas in conductors it attains large values and dominates the electromagnetic response.

The electric conductivity can be defined by the following methods:

Electric Conductivity Methods
Name	Description
Linear Isotropic	`ComplexElectricConductivityMethodLinearIsotropic` method specifies a constant electric conductivity $\sigma$.

Examples¶

The permeability, permittivity, and conductivity methods can be used in a following manner in order to define a general material for time-harmonic Maxwell model:

from mufem.electromagnetics.timedomainmagnetic import (
    ComplexMagneticPermeabilityMethodLinearIsotropic,
    ComplexElectricPermittivityMethodLinearIsotropic,
    ComplexElectricConductivityMethodLinearIsotropic,
    TimeHarmonicMaxwellGeneralMaterial,
)

# A typical quartz glass:
glass_permeability_method = ComplexMagneticPermeabilityMethodLinearIsotropic(1)
glass_permittivity_method = ComplexElectricPermittivityMethodLinearIsotropic(2.25)
glass_conductivity_method = ComplexElectricConductivityMethodLinearIsotropic(1e-15)

glass_material = TimeHarmonicMaxwellGeneralMaterial(
   name = "Glass",
   marker = glass_material_marker,
   permeability_method = glass_permeability_method,
   permittivity_method = glass_permittivity_method,
   conductivity_method = glass_conductivity_method,
)

# A typical metal:
metal_permeability_method = ComplexMagneticPermeabilityMethodLinearIsotropic(1)
metal_permittivity_method = ComplexElectricPermittivityMethodLinearIsotropic(100)
metal_conductivity_method = ComplexElectricConductivityMethodLinearIsotropic(1e7)

metal_material = TimeHarmonicMaxwellGeneralMaterial(
   name = "Metal",
   marker = metal_material_marker,
   permeability_method = metal_permeability_method,
   permittivity_method = metal_permittivity_method,
   conductivity_method = metal_conductivity_method,
)