The Ampère’s Law node adds Ampère’s law for the magnetic field to a domain and provides an interface for defining the constitutive relation and its associated properties as well as electric properties. There are two types of Ampère’s Law available; Ampère’s Law in Solids and Ampère’s Law in Fluids. This distinction decides how materials behave and how material properties are interpreted when the mesh is deformed.

Ampère’s Law in Solids applies to materials whose properties change as functions of material strain, material orientation, and other variables evaluated in a material reference configuration (material frame).

Ampère’s Law in Fluids applies to materials whose properties are defined only as functions of the current local state at each point in the spatial frame, and for which no unique material reference configuration can be defined.

Specify the constitutive relation that describes the macroscopic properties of the medium (relating the magnetic flux density B and the magnetic field H) and the applicable material properties, such as the relative permeability.

The equation for the selected constitutive relation displays under the list. For all options, the default uses values From material, or select User defined to enter a different value or expression.

Select a Magnetization model — Relative permeability (the default), B-H curve, Magnetic losses, Remanent flux density, Magnetization, or Effective B-H curve.

Select Relative permeability μr (dimensionless) to use the constitutive relation B = μ0μrH. For User defined select Isotropic, Diagonal, Symmetric, or Full and enter values or expressions in the field or matrix.

Select B-H curve ||H|| (SI unit: A/m) to use a curve that relates magnetic flux density B and the magnetic field H as ||H|| = f(||B||). Note that in literature B is typically given as a function of H, as ||B|| = f(||H||) or, when preserving direction: B = f(||H||)H/||H||. For the Ampère’s Law feature, however, B is closely related to the degree of freedom A and then H is derived from that.

The Magnetic field norm and Magnetic coenergy density settings can take the values From material or User defined.

When User defined is selected, specify a user-defined expression for the magnetic field norm. The direction of the magnetic field is taken to be the same as the direction of the magnetic flux density at each point.

Select Magnetic losses μ′ and μ″ (dimensionless) to describe the relative permeability as a complex-valued quantity: μr = μ′ − iμ″, where μ′ and μ″ are the real and imaginary parts, respectively. Note that the time-harmonic Sign Convention requires a lossy material to have a positive material parameter μ″ (see Modeling Losses in the Frequency Domain).

Select Remanent flux density Br (SI unit: T) to use the constitutive relation B = μ0μrecH + Br, where μrec and Br are the recoil permeability and the remanent flux density respectively (the flux density when no magnetic field is present). The recoil permeability μrec is very similar to the relative permeability, and is valid as long as the magnet is subjected to normal operating conditions (it is only valid within the linear region close to the vertical axis H = 0). Br is given by taking the remanent flux density norm (typically, provided by the material) and multiplying it with a normalized direction field specified in the physics: Br = ||Br|| e/||e||.

Select Magnetization M (SI unit: A/m) to use the constitutive relation B = μ0H + μ0M. Enter x and y components.

Select Effective B-H curve |H|eff (SI unit: A/m) to use an effective curve that provides the local linearized relation between the magnetic flux density B and the magnetic field H in time-harmonic problems.

Select External material to use a curve that relates magnetic flux density B and the magnetic field H as |H| = f(|B|) according to an externally coded function.

Specify the External material to use (from the Materials node under Global Definitions). This setting allows using material models or constitutive relations defined in an external library. See Working with External Materials for more information.

This section is described for the Current Conservation feature.

The default Relative permittivity εr (dimensionless) for the media is used From material and defined on the shell domain. For User defined, select Isotropic, Diagonal, Symmetric, or Full based on the characteristics of the permittivity and then enter values or expressions in the field or matrix.