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

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Basic Examples  (3)

Set up an electric flux density boundary condition:

Set up a surface charge density at the boundary:

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

Applications  (2)

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

Set up a region :

Specify electrostatic model parameter relative permittivity :

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

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

Specify the equation:

Solve the PDE:

Compute the electric field intensity E:

Visualize the solution:

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:

Set up a region :

Specify electrostatic model parameter relative permittivity :

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

Set up a ground potential at the lower boundary:

Specify the equation:

Solve the PDE:

Visualize the solution:

Wolfram Research (2023), ElectricFluxDensityValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html.

Text

Wolfram Research (2023), ElectricFluxDensityValue, Wolfram Language function, https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html.

CMS

Wolfram Language. 2023. "ElectricFluxDensityValue." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html.

APA

Wolfram Language. (2023). ElectricFluxDensityValue. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html

BibTeX

@misc{reference.wolfram_2024_electricfluxdensityvalue, author="Wolfram Research", title="{ElectricFluxDensityValue}", year="2023", howpublished="\url{https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html}", note=[Accessed: 27-April-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_electricfluxdensityvalue, organization={Wolfram Research}, title={ElectricFluxDensityValue}, year={2023}, url={https://reference.wolfram.com/language/ref/ElectricFluxDensityValue.html}, note=[Accessed: 27-April-2024 ]}