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MagneticFluxDensityValue[pred,vars,pars]

represents a magnetic flux density boundary condition for PDEs with predicate pred indicating where it applies, with model variables vars and global parameters pars.

MagneticFluxDensityValue[pred,vars,pars,lkey]

represents a magnetic flux density boundary condition with local parameters specified in pars[lkey].

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MagneticFluxDensityValue

MagneticFluxDensityValue[pred,vars,pars]

represents a magnetic flux density boundary condition for PDEs with predicate pred indicating where it applies, with model variables vars and global parameters pars.

MagneticFluxDensityValue[pred,vars,pars,lkey]

represents a magnetic flux density boundary condition with local parameters specified in pars[lkey].

Details

  • MagneticFluxDensityValue specifies a Neumann boundary value for the MagnetostaticPDEComponent.
  • MagneticFluxDensityValue specifies a boundary condition for the MagnetostaticPDEComponent and is used as part of the modeling equation:
  • MagneticFluxDensityValue is typically used to model a magnetic flux density field in units of [TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 2}}, webers per meter squared, {{(, "Webers", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]] that enters or leaves a boundary.
  • A positive value denotes the inward magnetic flux, and a negative value denotes an outward flux.
  • MagneticFluxDensityValue models a magnetic flux density , normal to the boundary with dependent variable in volts [TemplateBox[{InterpretationBox[, 1], "A", amperes, "Amperes"}, QuantityTF]] and independent variables in [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]].
  • Stationary variables vars are vars={Vm[x1,…,xn],{x1,…,xn}}.
  • The linear form of MagnetostaticPDEComponent with vacuum permeability in units of [TemplateBox[{InterpretationBox[, 1], {"H", , "/", , "m"}, henries per meter, {{(, "Henries", )}, /, {(, "Meters", )}}}, QuantityTF]] and relative permeability is given by:
  • MagneticFluxDensityValue with a specified scalar magnetic flux density , normal to the boundary, models:
  • MagneticFluxDensityValue models a specified magnetic flux density vector [TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 2}}, webers per meter squared, {{(, "Webers", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]] with unit normal :
  • Model parameters pars are specified as for MagnetostaticPDEComponent.
  • The following additional model parameters pars can be given:
  • parameterdefaultsymbol
    "BoundaryUnitNormal"Automatic
    "MagneticFluxDensity"{0,...}, magnetic flux density in [TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 2}}, webers per meter squared, {{(, "Webers", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]]
    "NormalMagneticFluxDensity", normal magnetic flux density in [TemplateBox[{InterpretationBox[, 1], {"Wb", , "/", , {"m", ^, 2}}, webers per meter squared, {{(, "Webers", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]]
    "Thickness", thickness in [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]]
  • All model parameters may depend on the spatial variables .
  • In two dimensions, the parameter "Thickness" is taken into account.
  • To localize model parameters, a key lkey can be specified, and values from association pars[lkey] are used for model parameters.
  • MagneticFluxDensityValue evaluates to a NeumannValue.
  • The boundary predicate pred can be specified as in NeumannValue.
  • If the MagneticFluxDensityValue depends on parameters that are specified in the association pars as …,keypi…,pivi,…, the parameters are replaced with .

Examples

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Basic Examples  (3)

Set up a magnetic flux density boundary condition:

Wolfram Language code: MagneticFluxDensityValue[{x, y}∈Subscript[Γ, b], {Subscript[V, m][x, y], {x, y}}, <|"MagneticFluxDensity" -> {B0x, B0y}|>]

Set up a normal magnetic flux density at the boundary:

Wolfram Language code: MagneticFluxDensityValue[{x, y}∈Subscript[Γ, b], {Subscript[V, m][x, y], {x, y}}, <|"NormalMagneticFluxDensity" -> Subscript[B, n]|>]

Set up a normal magnetic flux density at a boundary for a 2D magnetostatic model that has a thickness :

Wolfram Language code: MagneticFluxDensityValue[{x, y}∈Subscript[Γ, b], {V[x, y], {x, y}}, <|"Thickness" -> d|>, <|"NormalMagneticFluxDensity" -> Subscript[B, n]|>]

Scope  (2)

Create a magnetic flux density boundary condition with a boundary unit normal :

Wolfram Language code: MagneticFluxDensityValue[x == 0, {Subscript[V, m][x, y, z], {x, y, z}}, <|"MagneticFluxDensity" -> {10, 0, 0}, "BoundaryUnitNormal" -> {-1, 0, 0}|>]

Create a normal magnetic flux density boundary condition:

Wolfram Language code: MagneticFluxDensityValue[y == 0, {Subscript[V, m][x, y, z], {x, y, z}}, <| "RelativePermeability" -> Subscript[μ, r], "VacuumPermeability" -> Subscript[μ, 0]|>, <|"NormalMagneticFluxDensity" -> Subscript[B, n]|>]

Applications  (1)

Model an iron cube embedded inside air and emerged in a homogeneous magnetic field of [TemplateBox[{InterpretationBox[, 1], "T", teslas, "Teslas"}, QuantityTF]] directed along the axis. The domain is composed of an iron cube of length [TemplateBox[{InterpretationBox[, 1], "cm", centimeters, "Centimeters"}, QuantityTF]]. Due to symmetry, only 1/8 of the whole domain is simulated. The air boundary surrounding the iron cube is modeled as a second cube of length [TemplateBox[{InterpretationBox[, 1], "cm", centimeters, "Centimeters"}, QuantityTF]].

Define the mesh:

Wolfram Language code: mesh = \!\(\*Graphics3DBox[«6»]\);

The mesh has internal boundaries that represent the inner iron cube.

Define the iron cube:

Wolfram Language code: ironCube = Cuboid[{0, 0, 0}, {0.02, 0.02, 0.02}];

Visualize a wireframe of the mesh with the iron cube inside:

Wolfram Language code: Show[HighlightMesh[RegionBoundary[mesh], {}, PlotTheme -> "Lines"], Graphics3D[{Gray, ironCube}]]

Define variables:

Wolfram Language code: vars = {Vm[x, y, z], {x, y, z}};

The parameters to be used are the permeability of vacuum and iron .

Define parameters:

Wolfram Language code: pars = <|"RelativePermeability" -> Piecewise[{{1000.0, RegionMember[ironCube][{x, y, z}]}}, 1] * IdentityMatrix[3]|>;

To specify the homogeneous magnetic field across the domain, an outward magnetic flux density normal to the boundary at is specified.

Set up the magnetic flux density condition:

Wolfram Language code: fluxDensity = 1; Subscript[Γ, n] = MagneticFluxDensityValue[z == 0.1, vars, pars, <|"NormalMagneticFluxDensity" -> -fluxDensity|>]

Solve the magnetostatic PDE model:

Wolfram Language code: VmFun = NDSolveValue[{MagnetostaticPDEComponent[vars, pars] == Subscript[Γ, n], MagneticPotentialCondition[z == 0, vars, pars, <||>]}, Vm, {x, y, z}∈mesh]

Compute the magnetic field intensity:

Wolfram Language code: HField = -Grad[VmFun[x, y, z], {x, y, z}];

Visualize the field:

Wolfram Language code: Show[Graphics3D[{{Gray, ironCube}}, Boxed -> False], VectorPlot3D[HField, {x, y, z}∈mesh]]
Wolfram Research (2025), MagneticFluxDensityValue, Wolfram Language function, https://reference.wolfram.com/language/ref/MagneticFluxDensityValue.html.

Text

Wolfram Research (2025), MagneticFluxDensityValue, Wolfram Language function, https://reference.wolfram.com/language/ref/MagneticFluxDensityValue.html.

CMS

Wolfram Language. 2025. "MagneticFluxDensityValue." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/MagneticFluxDensityValue.html.

APA

Wolfram Language. (2025). MagneticFluxDensityValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MagneticFluxDensityValue.html

BibTeX

@misc{reference.wolfram_2026_magneticfluxdensityvalue, author="Wolfram Research", title="{MagneticFluxDensityValue}", year="2025", howpublished="\url{https://reference.wolfram.com/language/ref/MagneticFluxDensityValue.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_magneticfluxdensityvalue, organization={Wolfram Research}, title={MagneticFluxDensityValue}, year={2025}, url={https://reference.wolfram.com/language/ref/MagneticFluxDensityValue.html}, note=[Accessed: 12-June-2026]}