Assuming that we have obtained an image of the field, one would want
to know why field lines behave the way they do. Take the field generated
by two charges of opposite signs and different magnitudes q1 and q2; we
can see that:
An integrated software environment has been produced and used for the study of electrostatic fields created by point charges. The program allows modeling and visualizing arbitrarily complex systems of charges in a plane. Results include: geometric description and classification of field lines; methods that can be employed for arbitrary fields; sample field images, including all possible two-charge configurations. For all simple systems rigorous mathematical expressions are given. A correlation between electric flux and aspect of field lines is established.
Assuming that we have obtained an image of the field, one would want
to know why field lines behave the way they do. Take the field generated
by two charges of opposite signs and different magnitudes q1 and q2; we
can see that:
These aspects are discussed in section 2.
Field expert is a Windows application that can shed light over complex systems of charges. All images in this paper have been created with Field expert. The program takes the form of a specialized integrated environment and is designed to provide maximum ease of use:
Additional options:
The electric field E and the potential V created by a system
of point charges in any point is found by superposition:
where r is the position vector of the point and ri are the
position vectors of the charges.
Lines of force satisfy the differential equation: 
,
where Ex, Ey and Ez are the components of the vector E, of the form:
In the general case, there is no symbolic solution for this equation, so have to use numerical integration; however, as we will prove, symbolic solutions can be found for a system of point charges distributed on a straight line.
Equipotential surfaces satisfy the implicit equation 
Field lines are normal to equipotential surfaces in any point; also, since field lines are solutions of a differential equation, they don't cross.
Electric field and electric potential are related by
For a point charge q, the electric flux through a surface S is 
The negative sign is for surfaces where the flux is directed inwards
In this case it is possible to choose a simple coordinate frame:
The electric field is given by 
and the electric potential by 
.
Because we are in a bidimensional world, we will have equipotential
lines instead of surfaces.
The equation of lines of force contains now a single equality (because
the third coordinate z is not of concern) and can be re-arranged as:
.
This equations turns out to have the solution
For different values of the parameter C the equation describes all the
field lines. The physical meaning of this expression will be given later.
For y=0, 
The two factors that multiply q1 and q2 are precisely
Here a1 and a2
are the angles that describe the polar position of the target point relative
to q1 and q2. Consequently, the two factors should be between -1 and +1,
so that the parameter C must satisfy the inequation:
.
For other C's, the equation has no solution (the parameter does not describe
any line of force).
Inequation (2) is necessary, but not sufficient. The angles a1
and a2 must satisfy 
,
because APB is a triangle (a degenerated one maybe). It can be shown that
extremal values for C are obtained for points of AB; therefore the two
limits of C are the minimum and maximum value in the set { - q1 - q2 ,
q1 - q2, q1 + q2 }
a) If the two charges have like signs, the two extremal values for C are obtained
b) If the two charges have opposite signs, one limit will be on segment AB.
Looking at the images in the first paragraph one can see that
The question answered here is: what are the lines of force that pass through a given charge ? Refer to equation (1) again: since lines of force are controlled by the parameter C, we have to find a range for C.
For a line of force that passes through charge qk , the restriction of function f to that path is continuous in the neighborhood of the point (xk, 0), where qk lies; consequently, on that path the limit.
or, substituting
Take k=1 (for the other charge the results will be similar). Then xk =
-a; the second term poses no problem and can be taken out of the limit,
replacing x by -a and y by 0:
Remembering that the expression under the limit is a cosine, we get the
inequality 
, which
can be broken into 
.
At least one of these conditions will be satisfied automatically (it will
be equivalent to one of the inequalities in the validity inequation (2)).
The conditions for lines of force to pass through q2 are analogous:
The inequalities may or may not be strict: taking the equal sign, some
of them yield to reunions of singular paths.
To study the field lines further, we have to examine each case separately.
a) charges of same sign and different magnitudes: q1 > 0, q2 > 0, |q1| < |q2|.
The conditions become:
The separation between the two families is 
.
For y=0, 
So segment AB is part of the limit curve; the other part is a curve which passes between q1 and q2. Since it only crosses line AB once (within segment AB) and since field lines are infinite, the separation is an infinite curve, bending to the side of q1 (the smallest charge in terms of magnitude).
The two parts of the separation cross where the differential equation
has a singular point, that is at the point where E=0. From the equation
E=0 we can compute the position of this singular point: 
(closer to the smallest charge)
b) charges of opposite signs and different magnitudes: q1 > 0, q2 < 0, |q1| < |q2|
(C must be "large enough")
(all field lines pass through the charge of greatest magnitude)The separation is given by 
;
for y = 0, 
which is the ray starting at B and running away from A. The other part
of the separation is a bounded curve enclosing both charges (because
all field lines passing through q1 are bounded). The two parts cross at
a point where E=0
(outside segment
AB, closer to A, where the smallest charge lies)
c) charges of same sign and same magnitudes, q1 = q2 = q > 0
The configuration is symmetric about the vertical axis.
,
which is segment AB and the perpendicular through its midpoint O. In ,
E becomes zero.d) charges of opposite signs and same magnitudes
q1 = q > 0, q2 = -q < 0
The equation V(x,y) = is already in non-local form and will have as
solution a curve in the xy plane (the intersection of a surface with the
xy plane). The corresponding differential equations is: 
or, remembering that the partial derivatives of potential V are the components
of vector E:
and simply means that the normal to equipotentials in any point is the
electric field in that point.
Equipotentials are closed paths; some equipotentials close around one charge, some around more charges, and some enclose no charge at all (e.g.: four equal charges in the corners of a square). This can be a criterion for distinguishing families of equipotentials.
Given a point in the plane or a value of the potential V, it is difficult
to predict the behavior of the corresponding equipotentials. However, the
separation line between different families can be found using differential
geometry considerations. Equipotentials are plane curves; the separation
lines are given by those curves that have singular points (if an equipotential
has no singular point, it is a "regular" one, not a limiting
curve). The singular points of a curve are those for which all partial
derivatives become zero simultaneously; but, since the partial derivatives
of the potential V are the components of the electric field E, singular
points are those for which E=0:
All we have to do is
a) charges of same sign and different magnitudes: q1 > 0, q2 >
0, |q1| < |q2|.
The singular point has been found to line on the horizontal axis at
The potential at that point (the separation potential) is
The equipotential corresponding to Vs consists of two curves intersecting at (xs , y). For lower potentials, the equipotentials enclose both charges; for higher potentials, the equipotentials consist of isolated curves. Equipotentials that encircle Vs tend to a round shape, as if the system of charges were acting as a single charge. Thus, Vs can be used as a threshold when approximating the system with a single charge.
b) charges of opposite signs and different magnitudes: q1 > 0, q2
< 0, |q1| < |q2|
The singular point has
The potential at that point (the separation potential) is
The image differs from the previous case; one has to consider the "zero" equipotential V=0, which turns out to be a circle. Then
c) charges of same sign and same magnitudes, q1 = q2 = q > 0
This is just a particular case of (a). The separation potential is 
d) charges of opposite signs and same magnitudes q1 = q > 0, q2 =
-q < 0
This is the electric dipole case. There is a fundamental difference
from case (b) : there is no singular point and no separation potential.
The dipole can not be approximated by a point charge, no matter how far
we are, because no equipotential surrounds both charges.
The "zero" equipotential (the vertical axis) is an axis of (geometric) symmetry for the equipotentials. On the left, equipotentials correspond to positive values; on the right, to negative potentials.
Consider n charges on the horizontal axis, with their coordinates xk and their charges qk. The equation of the lines of force is similar to (1):
This equation can be deduced delimiting a "tube of force"
by revolving a portion of a line of force around the horizontal axis. Obviously
there is no flux through the "walls" of the tube. Then equation
(1') is really conservation of flux through any vertical section of the
"tube": flux is proportional to solid angle for point charges,
and the solid angle through a transverse section is 
.
Thus the methods designed above to study lines of force and equipotentials created by two charges can be applied to this situation as well.
For arbitrary systems of charges it is not possible to integrate the differential equations of a line of force. Equipotentials always have non-local equations (from definition), but then it is difficult to say anything about their shape in the general case; the conclusion that points where E = 0 lead are centers of separation between families of equipotentials holds, of course. One can easily get an overall picture of the field with the aid of Field Expert, write a qualitative description, and then focus on "hot points" to compute potential and field strength rigorously. Section 2.1 proves the effectiveness of this method.
The general behavior of the field is the following:
Exceptions:
NOTE: multipole configurations are rather hard to obtain, requiring very precise geometry; the quadripole may be the easiest -- four charges of equal magnitude and alternating signs in the corners of a square.
Another aspect from which I have shied away up to now is that the images produced by Field Expert are intriguing and, well, beautiful. I take the freedom to include some examples.
This simulates the field inside a plane capacitor; as expected, the field is almost uniform. The system acts as a dipole. Note: the "plates" would be much closer to one another normally.
Many singular points, separation curves and families of lines
The total charge of the system is zero. The system as a whole acts as a dipole.
I would like to thank Prof. MIHAI IORDACHE and Prof. ANCA IONESCU from "Politehnica" University for their guidance and help and for the excellent materials provided. Without them this paper would not be quite so complete.
The author of the Field Expert pogrom and of this paper is Dan Alin Muresan, first year student in the same university. Contact the author at his e-mail address
danm@morris.pcnet.ro danm@lbi.sfos.ro
The program uses the Euler numerical integration method to generate lines of force; starting from a given point, the line of force is followed pixel by pixel in both directions until a charge is "hit" or it runs away to infinity. Equipotential surfaces are traced using their definition, because they are closed curves and errors in the Euler lines method, which tend to cumulate, would make them open ! The equipotential is assumed to be smooth (with no sharp turns). See source code for more details.
The program is written in C++. The Windows interface is based on the Borland C++ 4.0 Object Windows Library. Electrostatic field computations are implemented in separate C++ classes, so the code is reusable for other applications. The program has been tested extensively under Windows 3.1 and Windows '95.
Copyright: Dan Alin Muresan
May 9, 1996