ElectricFluxDensityValue

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`ElectricFluxDensityValue`

New in 14[Experimental]

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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.

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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 [] and independent variables in [].
Stationary variables vars are vars={Θ[x₁,…,x_n],{x₁,…,x_n}}.
The polarized form of ElectrostaticPDEComponent with vacuum permittivity in units of [], 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:

parameter	default	symbol
"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 …,keyp_i…,p_iv_i,…], the parameters are replaced with .

Examples

open allclose all

Basic Examples (3)Summary of the most common use cases

Set up an electric flux density boundary condition:

In[1]:=1

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-vw7olo

Out[1]=1

Set up a surface charge density at the boundary:

In[1]:=1

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-u9ksq

Out[1]=1

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

In[1]:=1

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-giqrx

Out[1]=1

Applications (2)Sample problems that can be solved with this function

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

In[1]:=1

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-l31rgn

Set up a region :

In[2]:=2

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-mb3sl1

Specify electrostatic model parameter relative permittivity :

In[3]:=3

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-twb86

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

In[4]:=4

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-lhua7

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

In[5]:=5

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-bpt5u

Out[5]=5

Specify the equation:

In[6]:=6

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-hk3stc

Solve the PDE:

In[7]:=7

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-vxinx8

Compute the electric field intensity E:

In[8]:=8

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-e9rgz6

Visualize the solution:

In[9]:=9

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https://wolfram.com/xid/0dbjl13pdqimsb20oq-dp1xps

Out[9]=9

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 :