[Phys-L] spherical cows in the ivory tower +- ill-posed problems in the real world

[John Denker <jsd@av8n.com>]

On 02/27/2013 12:20 AM, Bernard Cleyet wrote:

> What is the mag. field surrounding a large, DC current carrying, diameter wire with a non co-axial hole thru it.  For simplicity assume infinite length.

That was the entire statement of the problem.  I quote it verbatim.

Some people examined the problem under one set of rules, while other
people used another set of rules.  That's interesting.  There's
nothing wrong with having different games with different rules.
However, it is important to keep track of which game you are
playing.  If you're supposed to be playing football, don't try
to use your baseball, bat, and mitt.

At one extreme, we have what I call "ivory tower" rules.  On an exam,
everybody (students and teacher alike) assumes that the students can
barely handle the material, so every imaginable simplifying assumption
is a correct assumption.  In particular, the problem is implicitly
assumed to have every imaginable symmetry that is not explicitly 
excluded by the statement of the problem.

In other words, if the exam asks you to calculate the capacitance of
a cow, you start by assuming a spherical cow, because this is the
best-case assumption.  If you do otherwise, people think you are a
kook and a troublemaker, making things more complicated than they
need to be.

Meanwhile, at the opposite extreme, there are real-world industrial-
strength rules.  If somebody hands you an ill-posed problem, you
do *not* get to make simplifying assumptions.  You certainly don't
get to make best-case assumptions.  For example, if you are the 
pilot in command of an aircraft, you do *not* get to assume that 
the baggage compartment is empty.  You also do *not* get to assume
that it is full.  You have to check.  If you assume wrongly,
people are going to die.

Along the same lines, suppose you are in charge of navigation and
guidance for Apollo 11.  You do *not* get to assume that the moon
is spherical and/or homogeneous.  You have to put satellites in
low orbit around the moon and do many measurements to locate
mass concentrations that put wrinkles in the gravitational field.
If you don't do this accurately enough, it jeopardizes the mission
and jeopardizes the crew.

You do not assume the system has the highest possible symmetry
consistent with the statement of the problem.  Au contraire, you
systematically consider the *lowest* possible symmetry consistent
with the statement of the problem.  You don't do this because you
are a kook or a troublemaker.  You do this to protect the lives
of the crew.

Again:  In principle there is nothing wrong with having two sets
of rules.  However, if your student Pollyanna Q. Pangloss has
exclusively played by ivory tower rules for 20 years, and then
graduates and suddenly has to play by real-world rules, baaad
things are going to happen.

Suggestion:  As students move through the educational system,
they should at least occasionally be asked to play by real-
world rules.  They should at least be aware that real-world
rules exist, and are radically different from ivory-tower
rules.

Not the sole purpose, but a major /part/ of the purpose of the
educational system should be to prepare people to get along in 
the real world.

===============
The rest of this note is not particularly important.  It just
documents the fact that in the real world, a current-carrying
wire might be dramatically less symmetrical than ivory-tower
assumptions (ITAs) would have you believe.

ITA #0:  The situation is a highly symmetrical as it could possibly
 be, except for the one symmetry that is expressly disclaimed.

 This is a highly nontrivial assumption.  At the very least, it
 raises the assumption:  If the intent was to describe a highly
 symmetrical situation, why did they not just say so?  The rule
 is simple:  mean what you say, and say what you mean.

The following can be considered subsets of the big ITA #0:

ITA #1:  The wire is straight.

 This is a nontrivial assumption, because there are lots of things
 in this world that are very long but not at all straight.  The
 wire could be curly, like a helix.  It could zig-zag in a regular
 pattern.  It could zig-zag in a random walk.  The wires from the 
 power plant to my home do not run in a straight line, not even 
 approximately.

ITA #2:  The hole does not run side-to-side across the wire.

 This is a totally non-obvious assumption.  The statement said the hole 
 goes "thru" the wire, not along the wire.  The #1 dictionary definition 
 of "through" involves going in one side and out the other.  Also the 
 statement said the wire had a "large" diameter.  This makes it plausible 
 that somebody could drill a transverse hole.  In the real world I've 
 seen *lots* of bus bars with holes drilled in them.  Thirdly, the 
 statement expressly disclaimed the most obvious non-transverse 
 arrangement.

ITA #3:  The hole is infinitely long. 

 This is a typical "ivory tower" assumption.  However, anybody who has 
 experience with real metal rods knows that they very commonly have 
 "stringers" i.e. voids or non-metallic inclusions that are very long
 but not as long as the whole rod.  This is a consequence of the way 
 the rod is made, by drawing.  Something that starts out as a bubble
or a chunk of slag gets stretched by the drawing process.

 The statement invited us to consider "infinite length" which I 
 assumed referred to the length of the wire.  Depending on how you
 diagram the sentence, it's not at all obvious that the hole is also
 intended to be infinite.

ITA #4:  The hole extends from one end of the wire to the other end,
 far, far away.

 This assumption is even stronger than the previous one, because 
 two things that are both infinite need not line up.  For example, if 
 the wire extends along the interval (-∞, ∞) and the hole extends along 
 the interval [0, ∞) then they are both infinite but the hole does not 
 extend over the whole length.

ITA #5:  The hole is straight.

 This is entirely nontrivial, even if the original wire is straight.  If 
 you take a straight wire and twist it around its axis, the wire remains 
 straight, but any stringers become more-or-less helical.

 In particular, in the real world there is a verrrrry great deal of
 stranded wire.  This can be well approximated as being a large-diameter
 chunk of metal with a bunch of helical holes in it.

 Less commonly but still possibly in principle, one could have a straight
 wire with a zig-zag hole in it.

ITA #6:  The cross-section of the wire is round. 

 In the real world, I've seen plenty of bus bar that is not at all round.
 The statement speaks of "diameter" but by definition non-round things are 
 allowed to have a diameter.  The main requirement is that a diameter has
 to go through the middle.

ITA #7:  The cross section of the hole is round.  

 This is even more nontrivial than the previous assumption.  The wire is made 
 by pulling it through a die that makes it round /on the outside/.  There is 
 no reason to believe this makes the hole become round.