Re: [Phys-l] induced electric field

["LaMontagne, Bob" <RLAMONT@providence.edu>]

Neat! The Ex = y and Ey = -x solution certainly gives a constant curl. The part of this that has always eluded me is what to do if there is there is such a large region of uniform dB/dt that you can't see the boundaries. The boundaries for the Betatron are clear, but I'm still not comfortable with a more general case. I take it that your comment about the problem being underspecified means that I really do need to know the boundaries and that knowledge makes the solution specific regarding the direction of E at any point within the boundary.

Bob at PC

________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker [jsd@av8n.com]
Sent: Wednesday, November 18, 2009 6:25 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] induced electric field

On 11/18/2009 08:56 AM, LaMontagne, Bob wrote:

> Where is the axis of this azimuth? If the region of dB/dt is very
> large an symmetric, I don't see how there is a specific direction for
> E. You could integrate curl E around a path to get an "EMF" , but how
> do actually identify E or its direction at a particular point?


That's an interesting question.  It's a simple question,
but answering it properly requires a number of sophisticated
ideas.


1) Basic physics tells us that at each point, the E field
must have *some* definite direction.  We can measure it
using a test charge.

If a given situation is so underspecified that we have
a hard time predicting what the direction of E will be,
that's a problem with the specification, not a problem
with the laws of physics.

2) A specification that tells us only the distribution
of B and of dB/dt is underspecified.  The relevant Maxwell
equation is a homogeneous linear partial differential
equation, and will always require some sort of initial
conditions and/or boundary conditions and/or constants
of integration before we can pin down a unique solution.

3) In particular, consider the following example.
We use x and y as coordinates, and use i and j as
unit vectors.  If Moe says that E(x,y) = yi + xj
is "the" solution, and Joe says that
E(x,y) = yi + (x-x0)j is "the" solution, i.e. just
Moe's solution but centered at a different point,
then the difference between the two solutions is
E(x,y) = -x0 j.  This difference field is constant
in time and uniform in space.  It is something you
get from electrostatics, and the magnetodynamic
equation has got nothing to say about it.  It's
just a constant of integration as far as the
magnetodynamic problem is concerned.

4) I like symmetry.  Given the choice between a
symmetric solution and a not-so-symmetric solution,
I am going to choose the symmetric solution every
time.  If we eventually need to shift it hither
and/or yon by adding some electrostatic constant
of integration, that can always be done in a later
step.

5) The question was asked, off list, how I knew it
would work out that way.

You might think I just solved the Maxwell equation
and discovered that it had certain mathematical
properties ... sorta the way I explained it in item
3 above ... but that is not how the idea came to
me.

It is complicated to explain, because there are
about ten arguments that all lead to the same
conclusion, and probably all ten of those neurons
were firing in my brain at the same time.

One important part of the picture was Feynman
volume II figure 3-9 (on page 3-9), which Feynman
cribbed from some guys named Stokes and Ampère.
It shows how a big uniform field can be built
up from small cells of uniform field ... and
all the Amperian currents on the cell boundaries
cancel, except on the final outermost overall
boundary of the region.  This is how I knew
there could not possibly be any difficulty with
a square solenoid.

Another part of the story was the fact that
_linearity implies superposition_.  I remember
being taught that.  Somebody said it, just once,
but it was so obviously important that they
didn't need to say it more than once.  In any
case, I've solved the Maxwell equations and
similar equations so many times that I just
know the solution cannot possibly be unique.
Add a uniform field and differentiate;  the
additive field drops out.  Indeed, add any
curl-free field and take the curl; the additive
field drops out.

I also knew that IF there was an identifiable
place where the field was zero, THEN it would
be easy to calculate the field everywhere else,
by starting at the zero-point and integrating
outward, concentric ring by concentric ring.
This is a symmetric process that must yield
a symmetric result.

Another part of the story was that I saw in my
mind's eye this picture of the E field inside
a betatron:
  http://www.av8n.com/physics/non-grady.htm#fig-betatron
Having spent hundreds of hours inventing that
way of picturing the field, I'm not likely to
forget it anytime soon.

I didn't bother to actually solve the equation
until the next day, when you guys challenged me
on it.  Given the solution, it is easy to work
backwards and see that it has the properties I
claimed for it.  Still, however, I emphasize that
that is not how I came up with the ideas.  I knew
how the solution(s) must behave long before I
saw any explicit representation thereof.

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