Re: [Phys-L] Helmholtz coils

[John Denker <jsd@av8n.com>]

On 06/08/2017 11:15 AM, bernard cleyet wrote:

> the field is constant only to first? order of the distance from the
> center

It is a sign of reasonably good intuition to put the question
mark in there.  However, one can do even better.

On 05/11/2017 09:55 AM, antti.j.savinainen wrote:

> > he [Feynman] supervised or co-supervised about 30 doctoral theses
> > as documented in a recent Physics Today piece. 
> > http://physicstoday.scitation.org/do/10.1063/PT.5.9100/full/

As was hinted at in the Physics Today article, Feynman had an effect
that extended far beyond direct supervision, far beyond any first-order
interaction.  People far and wide wanted to emulate his style.

Here is one small step down that road:

     *Don't let a good symmetry go to waste.*

Consider the on-axis field of a single coil of radius R.  It must
have a Taylor series, as a function of z/R.  We don't need to write
down the series to know it exists.

Now, when we use /two/ coils, symmetrically arranged about the
origin, the first-order term must vanish by symmetry.  Forsooth,
all of the odd-order terms must vanish.  The function, whatever
it is, must be an even function ... assuming the engineering was
done properly.

The zeroth order term survives.  It is the term we wanted all along.

Now, let's assume that Prof. Helmholtz was not the village idiot.
Let's assume that he arranged the spacing so as to extirpate\1\ the
lowest-order nonideal term that the two-coil setup would have had,
namely the second-order term.  That means the lowest surviving
nonideality is fourth order.

Fourth order!  That's way better than first order.  That means that
even if you are off-center by quite a bit, say z/(R/2) = 1/2, i.e.
z/R = 1/4, the field will still be within a few percent of nominal.

Using symmetry (another symmetry!) and conservation of flux lines,
we know the dependence on x and y will follow a similar scaling law,
indeed with a smaller coefficient in front of the nonideal term.

I'm not saying that's exactly how RPF would have done it.  He
might have come up with something much cleverer.  But my point
remains, there is no way he would have overlooked the symmetry
argument.


Here's a secondary point, a pedagogical point:  The skills involved
here can be taught.  Nobody is born with these skills.  Imagine an
environment where people are not satisfied merely to find a workable
solution to any given problem;  they work like crazy to find an
/elegant/ solution.

This looks like a bit of a paradox, but it can be resolved:
  Q: Why would anybody work so hard to find the easy solution?
  A: Because once you understand the elegant, incisive thought
   process, it is easy to remember ... and you get to use it
   again and again.  Assume you're going to be in this business
   for the rest of your life.  You can expect to see additional
   situations that are exactly the same (i.e. lots more Helmholtz
   pairs) or *not* exactly the same (e.g. Maxwell pairs!!!).
   More generally, there are eleventy bajillion situations where
   you can use symmetry to your advantage, for analysis and/or
   for optimal design.




\1\ Note: I said /extirpate/ (not "cancel") because it's zero for
each coil separately.  You don't need to know much about the field
to know it must have an inflection point as a function of z/R.  On
the opposite side of the same coin:  You can't expect the second
coil to cancel an even-order term.  It only cancels odd-order terms.

You do /not/ make a Maxwell pair by reversing one of the currents
in a Helmholtz pair.  It's one thing to know that;  it is another
thing to understand why.