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Re: [Phys-l] induced electric field



On 11/24/2009 07:25 AM, Philip Keller wrote:
When I asked my original question, I thought that there WAS a
contradiction about the direction of the electric fields and so I
wondered if there was no such thing as a uniformly increasing
magnetic field. Now I see that there is no contradiction about the
field direction.

:-)

It's a good lesson for us. If this stuff is confusing
to us, think how confusing it must be for the students.

But I still have issues with the term "uniformly
increasing uniform field".

Suppose we are located in a uniform electric field. That means that
the experiments we perform to investigate that field should come out
the same when we repeat them, translated a little to the left or
right.

Quite so.

But in a "uniformly increasing uniform magnetic field", that is not
the case? There is always a "center" to such a field? And by
performing repeated experiments with test charges, you can find the
center of that field. So uniform in this setting does not mean what
I thought it did...

Agreed, you cannot ever have a uniform electromagnetic
field with a uniformly-increasing magnetic component.
The Maxwell equations forbid it. A uniform E component
would have no curl at all, not the curl required to go
along with the uniformly-increasing magnetic field.
Another way of saying the same thing is to talk about
endless flux tubes in spacetime:
http://www.av8n.com/physics/flux-tubes.htm

There is always a "center" to such a field?

In simple symmetric situations, I reckon the induced
E field "usually" has a well-defined center, i.e.
a point where the induced field is zero ... but I
this is not a reliable general result. For starters,
consider a magnetic field that is uniformly increasing
inside a C-shaped or L-shaped region or other highly
non-convex non-symmetrical region. There could easily
be no "center" within the region ... and trying to
define a "center" outside the region is probably
not worth the trouble.

As usual, in the previous paragraph we assume
the intent was for the B field to be zero outside
the given region. This is just a language issue.

On the other hand, we have to be careful, because
the induced E field will be definitely nonzero
outside the given region. This is one of the
mistakes I made over the weekend: I wasn't
sufficiently careful about the external E-field.

A simple energy argument suggestions that the
induced E-field should be more-or-less "balanced"
in the following sense: You won't have a huge
E field all in one direction with only a small
curl, because that would involve too much E^2 dV
field energy. So the required curl will be made
up of a little bit of E in one direction that
curls around to become E in another direction,
creating some approximate overall "balance"
among the various directions. (Sorry for the
vagueness in this paragraph. Anybody who wants
to make this argument more precise is welcome
to do so.)