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[Phys-L] Re: Odd EM problem



On 04/08/05 18:50, Roger Haar wrote:

We came across an odd EM problem involving the
electric field induced by time varying magnetic
field. If it makes you happy, let it be large or
infinite in exten. Let it point vertically down,
be uniform, and be increasing in magnitude.

That's called a betatron field.

For
any given loop in the horizontal plane it is easy
to calculate the induced EMF around the loop and
from that you think you could figure out the
electric field at any point on the loop and thus
one could calculate the electic field at any spot
for a given time.

But here is the problem, first consider a loop
in the horizontal plane with radius of 1 meter and
located 1 meter to the right of the origin ( the
center is at (1,0,0) and use this loop to find
the electric fiel at the origin. Next calcuate
the electric feld at the orign by consider a
similar loop but with its center at (-1,0,0). In
one case the electric field seems to point in the
positive y direction while in the other it points
in the negative y direction.

The answer is simple yet profound: you cannot
calculate the E field from the given information.
There is a gauge symmetry. Moe can choose one
gauge, and Joe can choose another, and there's
no way to label either choice right or wrong.

A related fact is that given the circulation around
a loop, you cannot infer the field at any point on
the loop without additional assumptions. Sometimes
you can arrange it so that the loop sits at a place
of symmetry, so that the field is everywhere
tangential, and in this special case you can infer
the field ... but not in general.

The way to think about all such problems is to
appeal directly to the Maxwell equation. I like
the Clifford algebra representation
del F = 4 pi J [1]
http://www.av8n.com/physics/maxwell-ga.htm#eq-max-exp
but you can get the same result with an equally-
negligible amount of work in the older div/grad/curl
representation.

The statement of the problem tells us that the RHS
of equation [1] is zero, and on the LHS the term
involving the time derivative of Bx is nonzero. By
looking for corresponding terms, we see that there
are two (and only two) other terms that can be of
interest:
the x derivative of Ey
minus
the y derivative of Ex
and all you know is that *some* weighted combination
of those two adds up to cancel the magnetic term.
That is *all* you can know from the given statement
of the problem.

A useful picture of the situation can be found at
http://www.av8n.com/physics/non-conservative.htm#fig-betatron

For the gauge and origin I have chosen, at the 12:00
position on the diagram the field points one way; at
the 3:00 position it points another way, etc. etc.
Note that the B field is uniform and translationally
invariant. You can check that the circulation around
any loop you care to draw on the diagram is translationally
invariant.
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