I would like to plot the electric field of a dipole using Maple V RII for the macintosh.
A simplified vector form for the electric field of a dipole is:
[2*cos(theta)/r^3,sin(theta)/r^3]
As a first attempt I tried the following:
> Edipole:=[2*cos(theta)/r^3,sin(theta)/r^3]; > fieldplot( Edipole,r=-10..10,theta=0..Pi,coords=polar); Error, (in plot/options2d) unknown or bad argument, coords = polar
That method obviously didn’t work and I couldn’t figure out how to make a plot directly from the polar coordinates.
Then I tried transforming to cartesian coordinates.
>Edipole:=subs({r=(x^2+y^2)^(1/2),theta=arctan(y/x)}, [2*cos(theta)/r^3,sin(theta)/r^3]); > fieldplot( Edipole,x=-10..10,y=-10..10);
This method produced a plot but it did not look like a typical textbook picture of a dipole’s electric field.
Then I tried to plot the gradient of the electric potential of a dipole
> Vdipole:=subs({r=(x^2+y^2)^(1/2),theta=arctan(y/x)},cos(theta)/r^2); > gradplot( Vdipole,x=-10..10,y=-10..10);
This method produced a plot which looked exactly like that of the previous method.
I also seem to remember getting ’division by zero' errors as well while trying to use field
plot for this particular problem. However, I cannot seem to reproduce such an error at this
moment.
I wonder how one generally deals with ’division by zero’ errors in plotting routines.
Further, I would appreciate any suggestions concerning plotting the electric field of a dipole.
It is corrected with Maple V.3. (U. Klein)
http://userwst1.fh-reutlingen.de/~komma/physth/feldli1.ms
(should be updated, commented and translated ;-)
Release 2! I’m using Release 4 right now, but I think my remarks still apply.
| A simplified vector form for the electric field of a dipole is: ...
Release 4 doesn’t produce an error message here, but the result is not correct.
| Then I tried transforming to cartesian coordinates. ...
Three problems here:
1) Since the arrows grow in size as r -> 0, you only get a few big arrows and the rest very
tiny. You might try it without the r^3 in the denominator, to at least get the directions if
not the magnitudes of the arrows.
2) theta = arctan(y/x) only works in the right half plane. Use the two-variable form
theta = arctan(y,x) which works in the whole plane.
3) That’s _not_ the electric field of a dipole! What you want (for the cross-section of a
three-dimensional dipole) is [ 3*cos(2*theta)-1, 3*sin(2*theta)]/r^3
| I also seem to remember getting ’division by zero’ errors as well ...
I’ve run into it just now in R4 as well. For example,
> fieldplot([0,0], x = -1 .. 1, y = -1 .. 1); Error, (in plots/fieldplot/arrowsf2d) division by zero
This is a bug.
| I wonder how one generally deals with ’division by zero’ errors ...
If it comes from arrows of 0 length, try to avoid having arrows of 0 length. If they occur only at a discrete set of points or on a curve, you might shift the x and/or y endpoints a bit to miss these points. Or cheat by adding a small nonzero vector (too small to actually be seen in the plot) to the field.
| Error, (in plot/options2d) unknown or bad argument, coords = polar
This option is available in R4.
| Then I tried transforming to cartesian coordinates. ...
Besides the transformation of coordinates you must transform the fieldvector:
E[x]=E[r]*cos(theta)-E[theta]*sin(theta), E[y]=E[r]*sin[theta]+E[theta]*cos[theta]
| I wonder how one generally deals with ’division by zero’ errors ...
There are at least two reasons for ’division by zero’.
1. The singularity at r=0: shift the plot ranges, so that this point is not evaluated in
the grid (example: -1.001..1.002) or add a small constant in the denominator
(1/(r^3+.0001)).
2. If terms in the vector occur, that cannot be evaluated, Maples fieldplot reacts with a ’division by zero’ (due to zero-step-size I guess).
Plotting electrostatic fields (fieldlines) is best done with numerical methods, since the analytical solutions are the exception. The ’few’ closed solutions can be obtained by multipole-expansion of the *potential*.