Re: visualizing a non-potential

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

I wrote:
>>Here's an example.  Clockwise = down, everywhere.
>>   http://www.monmouth.com/~jsd/physics/img48/nonpot.png
>>
>>Non-experts tend to think that everything that is
>>a function of position has to be a potential, but
>>it's just not true.
>>
>>Pick two points A and B and count how many steps
>>you go down along a path from A to B. The answer
>>depends on your choice of path.  For a potential,
>>the answer would be independent of path.

On 04/26/2003 09:04 PM, Bob LaMontagne wrote:
> Exactly how is this a function of position? There is no actual "down" or
> "up", just a visual illusion that implies it.

It greatly depends on what "this" refers to.

-- Certainly there is no _height_ that is a function
  of position.

++ On the other hand, a stationary electron sitting
  in a cyclotron feels a _force_ that is a function
  of position.  This force is perfectly well behaved,
  but it is !!not!! the gradient of any potential.

If you drag the electron along some path from point
A to point B, the energy change is perfectly well
defined, but it depends on the path. If this were a
potential, the energy would depend only on the
endpoints, independent of path.

==

I don't consider this an illusion.  I consider it
a non-deceptive way of portraying non-potentials
such as the electric field in a cyclotron.

It's not a perfect portrayal, because it portrays
steps whereas the real field is smooth.  The steps
are a crutch that I need in order to portray what's
going on.  If somebody has a way of portraying a
non-potential without steps, please let me know!

In the meantime, you can just imagine a sequence
of diagrams like this, where the steps become
more numerous, with progressively smaller risers
and progressively smaller treads between them.

Also my use of the word "down" (as in "clockwise =
down everywhere") is imperfect, but I don't know
how to do better.  Non-potentials are not easy to
portray, and not even easy to talk about.  But
they are 100% real physics.  There is a _locally_
valid notion of "downward step" and "upward step",
even though there is no notion of height above
sea-level.

> Please write the function
> you have in mind as some f(x,y,z) to clarify this

The force vector acting on the electron has components
proportional to
       Fx =  y/sqrt(x^2+y^2)
       Fy = -x/sqrt(x^2+y^2)
       Fz =  0

Note that the magnitude of F is constant everywhere,
and that F points clockwise everywhere.

You can easily verify that this F is not the gradient
of any potential.  (Hint: del /\ del Phi is zero
for any potential Phi.)