Re: What is the interaction between 2 Helmholtz coils and a conductor?

When you say "two Helholtz coils" I am supposing that you mean the two
coils of a Helmholtz pair.  A Helholtz coil arrangement normally consists
of two identical coils on a common axis, separated by a distance equal to
the radius of the coils.  It has been found that this arrangement provides
a region with a fairly uniform magnetic field, which is useful for many
kinds of testing and evaluation.  If you search the web for Helmholtz coils
you will find many examples, including two and three-axis coil
arrangements.  The current in the two coils goes in the same direction, so
the answer to your first question is that the force on a conductor placed
between them would be in the same direction for each coil considered
separately, and if the conductor is in the middle of the gap, the forces
would be equal.  Here is a link to a site that shows how to calculate the
force;
 http://www.netdenizen.com/emagnet/helmholtz/idealhelmholtz.htm

Hope this helps.