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Re: [Phys-L] amusing electrostatics exercise



I picture the original wire in the shape of a large square, say with a side length of 10m. In the middle of one side of the square is a power supply. There is a break in the wire there, forming two ends and each end is attached to a terminal of the power supply. We are investigating the magnetic field within a few centimeters of the other side of the square, call it segment S. The fat wire diameter is on the order of a centimeter or two. Because we are so far from the ends of wire segment S and we are so far from the other wire segments, the assumption is that the magnetic field is well approximated by treating wire segment S as an infinitely long wire and ignoring the contributions from the other segments. The wire itself, has a circular cross section that has a circular hole in it. The hole in question extends the entire length of the wire. It's center line is parallel to the centerline of the fat wire itself. I replace the fat wire with two skinny wires, everywhere parallel to each other, one running along the centerline of the fat wire and one running along the centerline of the hole. I replace the power supply with two power supplies. Where is the charge conservation problem? What picture do you have in mind involving a kind of hole that affects only a short segment of the original wire? I think such a hole would violate the spirit of the original question.

-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@phys-l.org] On Behalf Of John Denker
Sent: Wednesday, February 27, 2013 3:56 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] amusing electrostatics exercise

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.
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