Keywords:
dielectric, electrostatics, capacitor
This simulation and the Dielectric in Electromagnetics (dielectricInEM.sdf) demonstrate the differences between simulating dielectric materials with electrostatic and electromagnetic solvers. They also demonstrate some of the more general differences between electrostatic and electromagnetic simulations.
Both of these simulations represent the same physical system: a slab of dielectric between two metal plates. The simulation grid is one square meter with a .75m x .25m slab of dielectric centered between the plates. The electric potential and electric field are solved over the entire domain.
Electrostatics is an approximation of the full set of Maxwell’s equations. According to Faraday’s law,
In the electrostatic limit, fields change slowly. More precisely, \(\frac{\partial B}{\partial t} \approx 0\), so that any curling electric field is negligible compared to the full electric field. When curling electric fields can be neglected, the electric field is a conservative field and can be written as the gradient of a scalar function. This scalar function is the electric potential, or voltage, and satisfies Poisson’s equation
Electrostatics is appropriate if the shortest wavelength of interest, \(\lambda_{EM}\) is much longer than the length scale of the simulation \(L_{s}\). This criterion, \(\lambda_{EM} > L_s\) respects the “fields change slowly” heuristic. Divide both sides of the expression by the speed of light and we get that the period of an electromagnetic oscillation is longer than the time it takes light to cross the simulation domain. In this limit, any change to a field that occurs within a timestep will have time to propagate throughout the simulation domain within the same timestep.
As can be seen in Poisson’s equation, electrostatics does not respect relativity, since any change in \(\rho\) will instantaneously change the electric potential everywhere in the simulation. So, it is important to respect the \(\lambda_{EM} > L_s\) criterion when doing electrostatics, otherwise, the simulation might neglect some important physical effects.
This simulation can be run with a VSimEM, VSimMD, or VSimPD license.
The Dielectric in Electrostatics example is accessed from within VSimComposer by the following actions:
The resulting Setup Window is shown Fig. 222.
In this simulation, voltages of 10.0 V and 0.0 V are set on the upper and lower boundaries (respectively) with Dirichlet boundary conditions. Neumann boundaries are used on the left and right walls of the simulation domain. In vacuum, this would set up an electric field of 10 V/m pointing down.
A dielectric is introduced to the simulation using a spacetime function
dielectricSapphire
. Sapphire has a relative dielectric constant,
\(\epsilon_r = \frac{\epsilon_{sapphire}}{\epsilon_0},\) of 9.8.
Through the use of Heaviside functions, we set the relative permittivity
to be 9.8 in a rectangular region between \(x = \pm .375\) meters an
\(y = \pm .125\) meters and 1 everywhere else.
In the setup tree, the dielectricSapphire
function is set as the
relative permittivity under Field Dynamics → PoissonSolver.
To run the simulation:
After performing the above actions, the results can be visualized as follows:
line out settings!
Change the dielectric constant, or change the location and area of the dielectric. Add a sinusoidal voltage on the plates.