Re: [Phys-L] amusing electrostatics exercise

[John Denker <jsd@av8n.com>]

On 02/27/2013 01:06 PM, Jeffrey Schnick wrote:
> Let the current in the original wire be I.  Replace the original wire
> with two skinny (negligible diameter) wires, wire C and wire G.  Wire
> C lies along what was the centerline of the original fat wire with
> the hollow tube in it.  Wire G lies along what was the centerline of
> the hollow tube.  Wire C has a current of I*A/(A-a) in the same
> direction as that of the original current and wire G has a current
> I*a/(A-a) in the opposite direction, where A is what the
> cross-sectional area of the original wire without the hole would be
> and a is the cross-sectional area of the hole.  The ends of wire C
> are connected to one power supply.  The ends of wire G are connected
> to another power supply. 

How exactly are those connections routed?

> Both power supplies are in about the same
> location as the original power supply that was causing the current in
> the wire with the tube through it.  In that region of space that
> would be outside the surface of the original wire if it were still
> there, the magnetic field is, to a very good approximation, a t
> locations far from the power supplies, identical to the magnetic
> field due to the original current carrying wire.

That is very specific about every point *except* the most
important point:  *how* do you route the connections to little 
wire G?  I claim that depending on how you do that, you can get 
a huuuuge range of different results.

Special cases include:
  a) Routing the connections along the path of the original
   wire.  This violates the spirit of the original question,
   namely that the hole affected only a short part of the
   original wire.

  b) Routing a twisted pair up to the location of little wire
   G, untwisting it only at the last moment.  This completely
   nullifies the effect of little wire G, since we have a
   loop with negligible area.

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

Pedagogical remark:  This situation is a trap for the unwary.

Consider the following scenario:
  People are taught about circuit diagrams, and they are taught
  about Kirchhoff's laws.  In accordance with Kirchhoff's laws,
  the circuit diagram is invariant with respect to topological
  distortions.  It is an abstract graph, divorced from geometrical
  notions such as distance.

When you start building real circuits, you quickly learn that any
such scenario is a load of baloney.  Maxwell's equations are right,
and Kirchhoff's "laws" are not right.  Until about a month ago,
the wikipedia article on "Kirchhoff's laws" said that they were
"directly derived from Maxwell's equations" ... but that is just 
not true.

You can do the experiment:  Toss a loop of wire onto a non-metallic
table, so that the loop has an area of about a square meter.  Hook
up a high-impedance instrument such as an oscilloscope, preferable
the 10x probe of an oscilloscope.  Observe the voltage.  Re-arrange
the loop and observe what happens.

The results can be explained in terms of Maxwell's equations.  The
source terms include:
 -- The ballast in a fluorescent light fixture is typically not
  well shielded.
 -- Motors are typically not well shielded.
 -- et cetera.  In a typical teaching lab or industrial lab,
  there are plennnty of stray fields running around.

The point is: Circuit geometry matters.

Sometimes you can engineer things so that it doesn't matter very
much, especially for non-critical applications, but often this
involves considerable amounts of overkill.  OTOH it is also quite
common for people to get into serious trouble by assuming it
doesn't matter.  At this point they need to hire a physicist to
come in and bail them out.  This guarantees there will always be
jobs for physicists.