WOLFRAM

Wolfram Language & System Documentation Center

ElectricFluxDensityValue[pred,vars,pars]

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

ElectricFluxDensityValue[pred,vars,pars,lkey]

represents an electric flux density boundary condition with local parameters specified in pars[lkey].

Details
Details and Options Details and Options
Examples  
Basic Examples  
Applications  
See Also
Tech Notes
Related Guides
History
Cite this Page

ElectricFluxDensityValue

ElectricFluxDensityValue[pred,vars,pars]

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

ElectricFluxDensityValue[pred,vars,pars,lkey]

represents an electric flux density boundary condition with local parameters specified in pars[lkey].

Details

  • ElectricFluxDensityValue specifies a Neumann boundary value for the ElectrostaticPDEComponent
  • ElectricFluxDensityValue specifies a boundary condition for ElectrostaticPDEComponent and is used as part of the modeling equation:
  • ElectricFluxDensityValue is typically used to model an electric flux density field in units of [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]] that enters or leaves a boundary.
  • The flux is caused by a surface charge density in units of [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]] positioned at a boundary.
  • A positive value denotes the inward electric flux, and a negative value denotes an outward flux.
  • ElectricFluxDensityValue models an electric flux density field normal to the boundary with dependent variable in volts [TemplateBox[{InterpretationBox[, 1], "V", volts, "Volts"}, QuantityTF]] and independent variables in [TemplateBox[{InterpretationBox[, 1], "m", meters, "Meters"}, QuantityTF]].
  • Stationary variables vars are vars={Θ[x1,…,xn],{x1,…,xn}}.
  • The polarized form of ElectrostaticPDEComponent with vacuum permittivity in units of [TemplateBox[{InterpretationBox[, 1], {"F", , "/", , "m"}, farads per meter, {{(, "Farads", )}, /, {(, "Meters", )}}}, QuantityTF]], polarization vector in units of [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]] and volume charge density in units of [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 3}}, coulombs per meter cubed, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 3}, )}}}, QuantityTF]] is given by:
  • When specified at an outer boundary ElectricFluxDensityValue models:
  • is a surface charge density in units of [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]] at an outer or interior boundary, and is a surface electric flux density in units of [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]].
  • When specified at an interior boundary between two nonconducting materials with electric flux density fields and , ElectricFluxDensityValue models:
  • Model parameters pars are specified as for ElectrostaticPDEComponent.
  • The following additional model parameters pars can be given:
  • parameterdefaultsymbol
    "ElectricFluxDensity"{0,...}, bondary electric flux density in [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]]
    "BoundaryUnitNormal"Automatic
    "SurfaceChargeDensity", surface charge density in [TemplateBox[{InterpretationBox[, 1], {"C", , "/", , {"m", ^, 2}}, coulombs per meter squared, {{(, "Coulombs", )}, /, {(, {"Meters", ^, 2}, )}}}, QuantityTF]]
  • All model parameters may depend on the spatial variables .
  • To localize model parameters, a key lkey can be specified and values from association pars[lkey] are used for model parameters.
  • ElectricFluxDensityValue evaluates to a NeumannValue.
  • The boundary predicate pred can be specified as in NeumannValue.
  • If the ElectricFluxDensityValue depends on parameters that are specified in the association pars as …,keypi…,pivi,…, the parameters are replaced with .

Examples

open all close all

Basic Examples  (3)

Set up an electric flux density boundary condition:

Wolfram Language code: ElectricFluxDensityValue[{x, y}∈Subscript[Γ, b], {V[x, y], {x, y}}, <|"ElectricFluxDensity" -> {D0x, D0y}|>]

Set up a surface charge density at the boundary:

Wolfram Language code: ElectricFluxDensityValue[{x, y}∈Subscript[Γ, b], {V[x, y], {x, y}}, <|"SurfaceChargeDensity" -> Subscript[ρ, s]|>]

Set up a surface charge density at a boundary, for a 2D electrostatic model that has a thickness :

Wolfram Language code: ElectricFluxDensityValue[{x, y}∈Subscript[Γ, b], {V[x, y], {x, y}}, <|"Thickness" -> d|>, <|"SurfaceChargeDensity" -> Subscript[ρ, s]|>]

Applications  (2)

Model a parallel electric field with an electric flux density boundary. Set up the electrostatic model variables vars:

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

Set up a region :

Wolfram Language code: Ω = Rectangle[{-1, -1}, {1, 1}];

Specify electrostatic model parameter relative permittivity :

Wolfram Language code: pars = <|"RelativePermittivity" -> 1|>;

Set up a ground potential at the right boundary of the box:

Wolfram Language code: Subscript[Γ, ground] = ElectricPotentialCondition[x == 1, vars, pars];

Set up an inward electric flux density boundary condition at the left boundary of the box:

Wolfram Language code: Subscript[Γ, Flux] = ElectricFluxDensityValue[x == -1, vars, pars, <|"ElectricFluxDensity" -> {8.85*^-7, 0}|>]

Specify the equation:

Wolfram Language code: eqn = ElectrostaticPDEComponent[vars, pars] == Subscript[Γ, Flux];

Solve the PDE:

Wolfram Language code: Vfun = NDSolveValue[{eqn, Subscript[Γ, ground]}, V, {x, y}∈Ω];

Compute the electric field intensity E:

Wolfram Language code: Efield = -Grad[Vfun[x, y], {x, y}];

Visualize the solution:

Wolfram Language code: VectorPlot[Efield, {x, y}∈Ω, ...]

Instead of specifying a voltage difference in a capacitor, one can also specify a surface charge density in one of the plates of the capacitor. Set up the electrostatic model variable vars:

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

Set up a region :

Wolfram Language code: Ω = Rectangle[{-0.05, -0.02}, {0.05, 0.02}];

Specify electrostatic model parameter relative permittivity :

Wolfram Language code: pars = <|"RelativePermittivity" -> 3|>;

Set up a positive surface charge density at the upper boundary:

Wolfram Language code: Subscript[Γ, p] = ElectricFluxDensityValue[y == 0.02, vars, pars, <|"SurfaceChargeDensity" -> 1*^-9|>]

Set up a ground potential at the lower boundary:

Wolfram Language code: Subscript[Γ, ground] = ElectricPotentialCondition[y == -0.02, vars, pars];

Specify the equation:

Wolfram Language code: eqn = ElectrostaticPDEComponent[vars, pars] == Subscript[Γ, p];

Solve the PDE:

Wolfram Language code: Vfun = NDSolveValue[{eqn, Subscript[Γ, ground]}, V, {x, y}∈Ω];

Visualize the solution:

Wolfram Language code: DensityPlot[Vfun[x, y], {x, y}∈ Ω, ...]
Wolfram Research (2024), ElectricFluxDensityValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html.

Text

Wolfram Research (2024), ElectricFluxDensityValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html.

CMS

Wolfram Language. 2024. "ElectricFluxDensityValue." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html.

APA

Wolfram Language. (2024). ElectricFluxDensityValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html

BibTeX

@misc{reference.wolfram_2026_electricfluxdensityvalue, author="Wolfram Research", title="{ElectricFluxDensityValue}", year="2024", howpublished="\url{https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_electricfluxdensityvalue, organization={Wolfram Research}, title={ElectricFluxDensityValue}, year={2024}, url={https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html}, note=[Accessed: 12-June-2026]}