Re: Electric fields and points of stability

["John S. Denker" <jsd@MONMOUTH.COM>]

Gary Turner wrote:
> .... place charges on the vertices of a
> cube.  Now does an equilibrium point exist?  It seems that it should - at
> the center of the cube - but what happens to the field lines at that
> point?  (There is no additional dimension to remove them).

1) Perhaps there is some confusion between
 -- equilbrium, and
 -- stability.

They're not the same.
  http://www.monmouth.com/~jsd/how/htm/equilib.html

2) There are no field lines at the center of the cube.
There are no field lines at the center of the square, either.
(If there were, it wouldn't be equilibrium.)

3) The zero in the middle of the cube is a higher-order
zero, i.e. more derivatives vanish.

4) Although there is a conservation law for "lines of force"
(which have to do equilibrium, not stability), there is
no corresponding conservation law for "lines of stability".
The center of the cube is equally unstable along all axes.

> Does this violate the (textbook) rule that field lines start and end on
> charges?

No.  There would be quite an uproar if it did.

> [More food for thought - 3-D, 4-D visualizations.  The potential can be
> plotted as a 3-D surface for the 2-D case.  Local min/max correspond
> to "quasi-equilibria".  I'm struggling to visualize a 4-D surface for the 3-
> D potential.  I usually do this as a "stack" of 3-D surfaces, but I'm just
> not seeing this one.  Any suggestions?]

a) In D=2, you can visualize the potential by contour lines
!!in!! the D=2 plane.  In D=3, correspondingly, you can
visualize the potential by equipotential contours (shells)
!!in!! the D=3 space.

b) Some people (not all) find it helpful to practice
visualizing _temperature_ as a function of position in
D=3 space;  it's an example of a relatively-familiar
scalar field.

This posting is the position of the writer, not that of Coulomb, Volta,
or Maxwell.

This posting is the position of the writer, not that of SUNY-BSC, NAU or the AAPT.