Re: [Phys-L] circular definition of "success" .... was: standard DC circuits

[John Denker <jsd@av8n.com>]

Let's bear in mind that the recent discussion of "standard DC 
circuits" began in the context of high-school physics.

What's appropriate for the general run of high-school juniors
is not necessarily the same as for engineering and science majors
in the second year of the calculus-based college physics course.
A 16 year old kid grows up a lot in three or four years.  Not
to mention the selection effects.


On 11/28/2013 10:22 AM, Bruce Sherwood mentioned:

> the great difficulty of making
> educational measurements that will be taken seriously.

Yes, there is verrry great difficulty.

By way of background, we have positive examples that tell us what
is required in order to conduct meaningful experiments on human
subjects.  For starters, consider the drug trials that pharmaceutical
companies carry out.  These require:
 -- Elaborate controls, including randomized controls.
 -- Double blinding.
 -- Long-term longitudinal studies.
 -- et cetera.

This is very expensive, but they do it anyway, because they know
that if they were to cut any corners, the results would not be 
reliable.

Evaluating a teaching method is a similar problem, only harder!
One reason that it's harder is the lack of double blinding, or
even single blinding.  Students and everybody else can see what
textbook is being used.

Compared to a drug trial, a typical education experiment applies
far less resources to a harder problem.  IMHO it is reasonable 
to be skeptical of the results.  I am not surprised to find that
almost any idea I can think of /and its opposite/ have been "proved" 
in the education literature.

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

Let us briefly consider the following imaginable ways of handing
surface charges in connection with DC circuits:
  a) Saying nothing at all about surface charges.
  b) Spending a lot of time saying wrong things about surface
   charges.  Kiplingesque just-so stories, without the humor.
   Obliterating the distinction between charge and voltage.
  c) Quantitative precision with maximum detail and maximum
   rigor mortis.
  d) Non-detailed yet fully correct understanding of the bedrock
   principles.  Direct application to selected /very simple/ 
   examples.  Mentioning (but not pursuing) the plausible idea
   that the same principles apply in complicated situations.
   This includes the principle that charge is not (!) the same 
   thing as voltage, and other principles listed below.
  e) For more advanced classes, using numerical methods to produce 
   correct diagrams for students to look at.  Students then apply 
   bedrock principles to check the results, to make sure that the
   computed solution makes sense, to connect it with other things
   we know.

> I note that in this
> Phys-L discussion no one has been unduly impressed with the evidence I've
> presented, despite its relevance.

Relevant?     Yes.
Dispositive?  No.

The evidence I've seen is nowhere near sufficient to convince me
that option (b) is the only viable option.

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

Teaching is hard.  Always was, always will be.

Just because the right stuff is hard does not give us a license to
teach wrong stuff.  And in this case, the right stuff isn't even
very hard, compared to the effort that has already been expended 
in teaching the wrong stuff.

Correct electrostatic principles include the following:

0) For the vast majority of practical DC circuit problems, you
 don't need to know very much about the electromagnetic field.
 Engineers have long since devised standard circuit structures
 for which the black-box properties are well understood.  Unless
 you are sure you want to rip open the black box, stop reading
 now, or skip directly to item (7) below.

 If you really want the lurid details, here you go:

1) The Coulomb interaction is a /long range/ interaction.  This 
 is diametrically contrary to what the _M&I_ book implies on page
 383, but it is true nonetheless.  Formally the electrostatic field
 is an /infinite-range/ interaction, if you want to get technical
 about it.

 You can show students this familiar diagram:
  http://www.av8n.com/physics/img48/infinite-range-interaction.png
 and say something like this:  At 2X the distance, the force is 4X
 smaller (in accordance with Coulomb's law), but because of the 
 geometry it affects 4X as much stuff.  This is not a proof, but 
 it works as a plausibility argument.  This is important, because
 it means that a small amount of charge up close can be just as 
 significant as a large amount of charge far away -- and vice versa!

2) The electromagnetic field is /linear/.  That means that if 
 charge A produces field Ea and charge B produces field Eb, then
 if you have charges A and B together the total field at each
 location (r) in space can be found by simply adding the vectors
 Ea and Eb.  That is:
         Eab(r) = Ea(r) + Eb(r)                            [1]

 Draw the vector addition diagrams.

3) Item (2) guarantees that finding the charge distribution will
 be conceptually straightforward.  However, item (1) guarantees
 that it will be quite laborious, except in ultra-simple cases.
 This is just the sort of thing that computers handle really well. 
 There is nothing tricky going on, but there are a lot of moving 
 parts to keep track of.

4) It is easier to verify a solution than to construct it.  So 
 use a computer program to construct a solution, then check it
 N different ways.

5a) Simple example:   Infinitely long straight uniform nichrome 
 wire carrying a current.  

     >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
                (wire)

 We assert the following solution, and then check it:  A uniform E 
 field in all of space, including inside the wire.  No surface 
 charges anywhere on the local piece of wire.  Ohm's law is satisfied.
 The laws of electrostatics are satisfied.  You can easily check all 
 this.

5b) Next example:  Same as above, except that there is an additional
 electrode that messes up the field.

               (electrode)
                   XX


     >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
                (wire)

 At this stage of the game, we have a setup that does not satisfy 
 the DC equations of motion, but it can exist temporarily.  Draw
 the E-field vectors to show that the E-field plus Ohm's law
 produce a current that does not flow in the direction of the
 wire.

5c) A transient AC current will flow.  It will deposit charges on 
 the surface of the wire, very roughly as shown below:


               (electrode)
                   XX

                 - -- -
     >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
                (wire)

 The idea is that in the steady state, the surface charges screen 
 out -- i.e. cancel -- the field coming from the electrode, so 
 that the field /within/ the wire is the same as in scenario (5a). 
 Draw the vector-sum diagrams.

 It can be shown that so long as we have positive capacitance
 and positive resistance, the transient currents flow in the 
 correct direction, moving charge to where it is needed.  They
 keep flowing until the situation converges to the DC solution.
 The field in the wire is then the same as in scenario (5a).
 The field elsewhere is messy, but we don't care about that. 
 We care a lot about the field in the wire, because that's where
 the mobile charges are.

6) Charge is not the same thing as voltage.  This is a bedrock
 principle.

 The voltage in scenario (5a) decreases steadily as we move
 from left to right.  This absolutely does not mean there is
 a steadily decreasing positive charge density or increasing
 negative charge density.  Forsooth, there is no surface charge 
 along the length of the wire anywhere.

 The _M&I_ words and diagrams demand a charge density proportional
 to voltage.  I've been racking my brain trying to figure out where 
 this misconception could be coming from.  Here's the most plausible
 hypothesis I've been able to conjure up so far:  It would make 
 sense to have charge proportional to voltage if you thought the 
 wire had a constant capacitance per unit length.  However, alas,
 this notion is wrong several times over:
   a) An ideal wire does not have constant capacitance per unit 
    length except in certain carefully-engineered situations, 
    and the serpentine geometry in figure 19.14 is certainly not
    one of those situations.  In general, the capacitance of an
    ideal wire depends on both the diameter of the wire and how 
    close it is to other things.
   b) At a much more profound level, we don't have an ideal wire.
    The whole idea that a capacitor should have a charge proportional
    to the voltage rests on a crucial yet often-unstated assumption,
    namely that the capacitor plate is a good conductor!  The 
    following conditions apply to a good conductor:
    -- The plate surface is an equipotential.
    -- There are no field lines in the interior of the plate.
    -- All the electrical field lines in the vicinity are 
     perpendicular to the plate surface.

    If all three of those conditions were true, it would imply
    a surface charge density proportional to the field, since
    every field line must end on a charge.  For fixed geometry,
    this would imply charge proportional to voltage.

    However, none of those three conditions is valid when the
    nichrome wire is carrying a current.  So it looks like we 
    are getting bit by multiple misconceptions all at once.

    A great deal of your intuition about basic capacitors goes
    out the window when the plate is in a far-from-equilibrium
    situation such as this.  This is not a "basic" capacitor.

 Correct idea:  The charge that we added in going from (5b) 
 to (5c) produces a /change/ in the /field/, in accordance with 
 the superposition principle (which is another bedrock principle).

 The strategy of /incremental/ amounts of charge accumulating
 to make /incremental/ corrections to the field pattern can
 be extended to arbitrarily complicated circuit geometries.

    In contrast, it is a misconception to think that the charge 
    determines the change in voltage, or determines the field 
    itself (which is the gradient of the voltage).

        It is a double misconception to think that the charge
        in figure (5c) dictates the voltage directly.  This is 
        what the _M&I_ diagrams are trying to tell you, but it
        is quite wrong.

7) For linear circuits, calculating the fields and charges is 
 simple in principle.  Linearity to the rescue!  Still, it 
 gets tedious if the circuit is big and complicated.
  a) For arbitrary geometry, you can use a computer program 
   to find the voltages, fields, and charge distributions.
  b) Electrical engineers have long since developed standard
   geometries for which the macroscopic black-box properties 
   are well known.  In simple low-frequency circuits, "most"
   of the field energy is /inside/ the capacitors, inductors, 
   and mutual-inductances (aka transformers).  There is no 
   law of physics that requires this to be true, but suitable
   engineering can make it be true, to a good approximation,
   especially at low frequencies.  See item (0) above.

8) We are not violating any God-given commandments by making 
 engineering approximations when analyzing DC circuits.  After
 all, as a matter of principle, the very existence of a "DC"
 circuit is an approximation.  You can choose to talk about 
 the /behavior/ of a circuit in the DC limit, but that cannot 
 possibly be the only behavior.


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

A minor remark about the reference people have been citing:
  Rainer Mueller
  "A semiquantitative treatment of surface charges in DC circuits”
  American Journal of Physics 80 (9) Sept. 2012, pages 782-788.
   http://scitation.aip.org/content/aapt/journal/ajp/80/9/10.1119/1.4731722
   https://www.tu-braunschweig.de/Medien-DB/ifdn-physik/ajp000782.pdf

The field and charge distributions in that paper all seem to pertain 
to the 2D situation.  This makes them similar but not identical 
to my results, which pertain to a 3D situation.

3D eats up more CPU time, but if you aren't trying for super-
high resolution it isn't too bad.  It takes less than 20 seconds
on a beat-up old laptop to generate these figures:
  http://www.av8n.com/physics/img48/rwire_material.png
  http://www.av8n.com/physics/img48/rwire_charge.png

On 11/28/2013 10:22 AM, Bruce Sherwood wrote:

> I've addressed the issue of the crudeness of the charge
> distributions shown in our book, in particular the emphasis on field, with
> the charge distribution continually labeled as very approximate, and
> there's nothing I can add to that.

I suggest not giving up so easily.  There's lots more that could be
done.  At some point it is easier to revise the words and diagrams
so they are not such a breeding ground for misconceptions.  At some 
point this is easier than going to such great lengths to warn people
about the problems.

I insist that the caveats, "continual" though they may be, do not
solve the problem, because the caveats themselves are not correct.
The discussion on page 761 is clearly written.  It quite clearly
spells out an incorrect idea.  It identifies the "key idea" which 
is allegedly correct, and distinguishes it from the "very rough
approximations" that arise when applying the idea to complicated
bendy geometries.  Alas, the "key idea" is just not correct.  See
item (5) above.

There is no advantage in oversimplifying the diagram in the book.
If students look at the correct charge distribution, it will not 
cause their eyeballs to fall out of their heads.

Even if they cannot draw the correct charge distributions a_priori
by hand, it is still possible to understand a_posteriori what the 
correct diagram is saying:
 -- There tends to be a high concentration of charge at the external
  corners.  Check.  That makes sense.  That has direct application 
  in the real world, e.g. corona points.
 -- There tends to be a very low concentration of charge on the inside
  corners.  Check.  That makes sense.  That is consistent with what
  we know about real-world Faraday cages, et cetera.
 -- There are major concentrations of charge at the places where something
  at one voltage is geometrically near something at a different voltage.
  Check.  That is consistent with basic notions of capacitance.
 -- In the interior of a long straight wire, not too near the ends,
  there is a field and there is a current, but there is no net charge 
  density  Check.  That makes sense in terms of basic notions of 
  electrostatic repulsion.
 -- The correct charge distribution does *not* have charge density 
  proportional to voltage.
 -- et cetera..............

And (!) it is possible to draw the correct charge distribution by hand
in sufficiently-simple situations, as in item (5) above.

I utterly fail to see the advantage of feeding students wrong words and
wrong diagrams, let alone requiring them to draw wrong diagrams by hand.

Saying that the charge distribution is "approximate" doesn't help, because
virtually everything we do is approximate.  For example, the formula p=mv 
is approximate.  The smart approach is to make a /controlled/ approximation.
The p=mv approximation is well controlled, insofar as we understand quite
well how it relates to the exact answer.  In contrast, the idea of surface
charge proportional to voltage is just a terrible, terrible idea.  Saying
that the idea applies only "very roughly" to complicated circuits fails to
communicate how terrible it is.  The idea fails utterly and categorically 
in even the simplest situations, as we saw in item (5) above.  By way of 
analogy:
 -- Setting π=3 is a rough approximation.
 -- Setting π=0 is a travesty.

It is claimed that figure 19.17 has the advantage that it can be drawn 
by hand.  So what?  What is the advantage of a rapid way of getting the 
wrong answer?  What is the advantage of a convenient way of cultivating
a slew of misconceptions?

Again I say that the correct principles /can/ be investigated by hand
if need be, and can be understood just fine in simple cases.  Insisting 
on attacking the complicated cases by hand is a self-inflicted wound.

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

I am still fascinated by the "critical thinking" issues.  How many 
professors have taught this course?  How many students have taken 
this course?  Didn't any of them notice that the discussion on page 
761 is inconsistent with the bedrock principles of electrostatics?
Not just approximate as to details, but inconsistent as to the whole
premise.  I wouldn't expect all of them to notice, but I would have 
expected /some/ of them to notice.

As others have said, it might be good to take the metacognition bull
by the horns and /teach/ everybody how to notice things like this.