The contour plot of the electric
potential for a random distribution of charges near an air-water
interface is shown in figure 1. Spatial dimensions are in units
of Angstroms. Positive charges are shown in blue, negative
charges in red. All charges are ±1 electron charge.
Figure 1 - Equipotentials for
a random charge distribution near air-water interface
Figure 2 shows the field lines for this configuration. These
field lines were found by solving numerically solving the differential
equations of the electric field, using the Mathematica command NDSolve.
2 - Field lines for charges near air-water interface
Included in the notebook is alternative code for producing the field
lines - the method used to produce the field lines in figure 2 uses
NDSolve to compute the field lines from the parameterized differential
equations, with "t" the parameter. Since the electric field can
be relatively weak in materials of high dielectric constant, the
NDSolve routine must be run for quite some time to produce field lines
which actually run from one charge to another. Another problem
that occurs occasionally when charges of similar strength are located
near each other is that the initial step along the field line is too
large, which results in the field line making too large a jump in one
direction - this can lead to field lines crossing each other, which
obviously should not happen!
To counter this problem, I included code which plots the field lines
using a much simpler method, similar to the Euler method of solving
DE's. This method takes fixed spatial steps along the field
lines, rather than steps in the time parameter t. Figures 3 and 4
show a particular example of how this method may be more effective than
the more advanced NDSolve method. Note in figure 3 that some of
the field lines are quite rough, and some of the field lines
cross. In figure 4, which was produced using the simpler Euler
method, the field lines are much smoother.
Figure 3 - Field lines
plotted using NDSolve method - note the crossed field lines
Figure 4 - Field lines
plotted using simpler Euler method - much smoother field lines
I would recommend using the alternative method a majority of the time
for this problem, it seems to be much more reliable and doesn't take as
long to produce the field lines. However I would recommend the
user experiment with both methods and decide for themself.