Here we have a rectangular slab of variable dielectric constant, epsilon, inside a rectangle with epsilon=1
and the potential at the top of the top is +/-1 separated by a gap. This arrangement is often used for integrated optics chips
where an electric field near the surface of a substrate is needed to modify the phase of a wave in a waveguide near the surface.
When the charge density is zero, the differential equation for the potential, V, is a modified version of
the Laplace equation Divergence(epsilon*gradient(V))=0.
In order to efficiently solve for V(x,y) using finite difference methods we need to have a good estimate of V(y) for the internal nodes.
If you set the number of iterations to zero you can see the estimate being used here.
If the dielectric constant is the same everywhere,
then one can also use complex function mapping to plot the potential V(x,y)and that is an option available here.
For the finite difference method that I used to refine the guess, I very strongly recommend the following link:
https://atmos.washington.edu/~breth/classes/AM585/lect/rjl_585. Since we need the divergence of epsilon*gradient(V)
and epsilon(x,y) has x and y variation we have to take the first differentials of epsilon and V in addition to the
second differential of V: i.e. d/dx(eps*dV/dx)=eps*d2Vdx2+deps/dx*dV/dx and similarly for the y differential.
The electric field, -gradient(V), is inversely proportional the the spacing between the potential contours.
It is important to notice the way the dielectric slab greatly excludes the electric field.
This is similar to the way a superconductor excludes an external magnetic field.
Eplanation of the controls:
The dimension sliders have the name "Ratio" instead of simple lengths. The ratio is the ratio of the dimension to the
dimension (height or width) of the display rectangle.
The calculation of V(x,y) can be fairly slow so I have also provided another ratio, the calculation to display rectangle
ratio. To increase accuracy choose this ratio to be 1 and to increase speed choose the ratio to be less than 1.
Boundary Conditions:
I have chosen to have the normal component of the gradient(V) be zero on all boundaries except the electrodes where the
potential is specified. This is called Neumann boundary conditions and it means that all V(x,y) contours intersect the boundary.
at normal incidence. For the heat flow equation this is the condition where no heat flows out of or into the rectangular region.
For the electric potential equation it is the condition where the normal force on a test electron is zero at the boundaries.