ElectrostaticPDEComponent
ElectrostaticPDEComponent[vars,pars]
yields an electrostatic PDE term with variables vars and parameters pars.
Details
- ElectrostaticPDEComponent is typically used to generate an electrostatic equation with model variables vars and model parameters pars.
- ElectrostaticPDEComponent returns a sum of differential operators to be used as a part of partial differential equations:
- ElectrostaticPDEComponent models static electric fields produced by stationary charges in insulating, or dielectric materials.
- ElectrostaticPDEComponent models electrostatic phenomena with the dependent variable , the electric scalar potential. is in units of volt [], independent variables in units of [].
- Stationary variables vars are vars={V[x1,…,xn],{x1,…,xn}}.
- ElectrostaticPDEComponent generally does not produces a time-dependent PDE.
- ElectrostaticPDEComponent is based on a diffusion, a source and a derivative PDE term:
- is the vacuum permittivity in units of [], the polarization vector in units of [] and the volume charge density in units of [].
- The polarization vector specifies the density of permanent or induced electric dipole moments inside a material.
- The volume charge density models charge distributions, negative or positive.
- ElectrostaticPDEComponent can produce different equations, depending on the constitutive relationship.
- For linear materials, the ElectrostaticPDEComponent equation simplifies to:
- is the unitless relative permittivity.
- can be isotropic, orthotropic or anisotropic.
- For nonlinear non-hysteresis ferroelectric materials, the ElectrostaticPDEComponent equation is given as:
- is the remanent polarization vector in units of [].
- The implicit default boundary condition for the electrostatic model is a 0 ElectricFluxDensityValue.
- The units of the electrostatic model terms are in [], or equivalently in [].
- The following parameters pars can be given:
-
parameter default symbol "Polarization" {0,…} , polarization vector in [] "RegionSymmetry" None "RelativePermittivity" - , unitless relative permittivity
"RemanentPolarization" {0,…} , remanent polarization vector in [] "Thickness" 1 , thickness in [] "CrossSectionalArea" 1 , cross-sectional area in [] "VacuumPermittivity" , vacuum permittivity in [] "VolumeChargeDensity" 0 , volume charge density in [] - All parameters may depend on the spacial variable and dependent variable .
- The number of independent variables determines the dimensions of or and the length of vectors and .
- A possible choice for the parameter "RegionSymmetry" is "Axisymmetric".
- "Axisymmetric" region symmetry represents a truncated cylindrical coordinate system where the cylindrical coordinates are reduced by removing the angle variable as follows:
-
dimension reduction equation 1D 2D - In 2D, when a "Thickness" is specified, the ElectrostaticPDEComponent equation is given as:
- In 1D, when a "CrossSectionalArea" is specified, the ElectrostaticPDEComponent equation is given as:
- In a 1D axisymmetric, when a "Thickness" is specified, the ElectrostaticPDEComponent equation is given as:
- The input specification for the parameters is exactly the same as for their corresponding operator terms.
- If no parameters are specified, the default electrostatic PDE is:
- If the ElectrostaticPDEComponent depends on parameters that are specified in the association pars as …,keypi…,pivi,…], the parameters are replaced with .
Examples
open allclose allBasic Examples (3)
Scope (14)
1D (4)
Define a 1D electrostatic model with a cross-sectional area :
Model an electric potential field with two electric potential conditions at the sides.
Specify model variables and electrostatic parameters:
Set up and solve an electrostatic PDE:
Model an electric potential field with two electric potential conditions at the sides and a discontinuous relative permittivity.
Specify model variables and electrostatic parameters:
2D (5)
2D Axisymmetric (2)
3D (1)
Multi-material (2)
Applications (4)
1D (2)
Compute the electric potential distribution between two parallel plates separated by a distance [] and positioned normal to the axis. The left plate is maintained at a constant potential [], whereas the right plate is grounded, []. The region between the plates is characterized by a relative permittivity and a uniform electron charge density []. The equation to model is given by:
Set up the electrostatic model variables :
Specify electrostatic model parameters:
Specify the electric potential conditions:
Compute the electrical potential distribution between two parallel plates with the same distance and boundary conditions as in the previous example, but now with a nonuniform charge distribution given by:
Set up the electrostatic model variables :
Specify electrostatic model parameters:
2D (1)
Model an infinitely long rectangular box with metallic walls. The box has a width of [], and a height of []. The side and bottom walls are maintained at zero electric potential, whereas the top wall has a fixed electric potential of . The region inside the box is free of charge . The equation to model is given by:
Define the model variables and parameters:
Set up a 2D electrostatic model:
The side and bottom walls at , and are maintained at a zero electric potential:
3D (1)
Model a dielectric material of a cylindrical capacitor with two electric potential conditions at the upper and lower boundaries, which represent the capacitor electrodes. The equation to model is given by:
Set up the electrostatic model variables :
Specify a relative permittivity :
Specify ground potential at the lower boundary:
Possible Issues (1)
For symbolic computation, the "VacuumPermittivity" or "RelativePermittivity" parameter should be given as a matrix:
For numeric values, the "VacuumPermittivity" or "RelativePermittivity" parameter is automatically converted to a matrix of proper dimensions:
This automatic conversion is not possible for symbolic input:
Not providing the properly dimensioned matrix will result in an error:
Text
Wolfram Research (2024), ElectrostaticPDEComponent, Wolfram Language function, https://reference.wolfram.com/language/ref/ElectrostaticPDEComponent.html (updated 2024).
CMS
Wolfram Language. 2024. "ElectrostaticPDEComponent." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/ElectrostaticPDEComponent.html.
APA
Wolfram Language. (2024). ElectrostaticPDEComponent. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ElectrostaticPDEComponent.html