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# Re: equipotentials

• From: John Denker <jsd@AV8N.COM>
• Date: Sat, 21 Feb 2004 01:13:59 -0800

Quoting Justin Parke <FIZIX29@AOL.COM>:

Can equipotential surfaces cross? If so then in what direction does the
electric field point at the line of intersection?

(This is another disagreement between me and the "back of the book" answer
for a textbook question)

There is danger of this degenerating into word-games.

To prevent that, consider a physical example: four line
charges (equal in magnitude) arranged in the following pattern:

+ -

- +

By symmetry, the XZ plane is an equipotential. Also the
YZ plane is an equipotential These two planes cross in
the obvious way ... their intersection is the Z axis.

Now I would prefer to call these planes two different
branches of a *single* funny-shaped equipotential surface.
Calling them "two" equipotential surfaces would seem like
word-games to me.

===============

It is *usually* true that if two such branches cross or meet
at a point, the field vanishes at that point. You can make
an argument about linear independence or lack thereof. But
you need to put on some restrictions: for starters you
ought to ensure that the branches cross at a nonzero angle
(as opposed to just kissing).

======================

Of course the equipotential surfaces corresponding to two
different potential-values can never cross. By definition of
equipotential. Indeed this hinges on the definition of equal.