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Re: [Phys-l] induced electric field



Thank you for the reply -- I will need to read this all more slowly. But for now, I hope this is an easy question. What happens to an isolated point charge in a region in space with a uniformly increasing uniform mag field? Does it experience a force? If so, in what direction? Does it matter where I release it?



-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of John Denker
Sent: Tuesday, November 17, 2009 11:27 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] induced electric field

On 11/17/2009 06:14 AM, Philip Keller wrote:
Is it possible that there can be no such thing as a uniform magnetic
field that is uniformly increasing? The reason I ask is:

You _can_ have a field that is uniform over any finite
region -- even a very large region -- and it can be
uniformly increasing. A _betatron_ is a celebrated
example of a machine that uses such a field. For
details, see below.

Suppose there were such a region and you wanted to know the electric
field at a point in that region. This is a typical textbook problem:
if there were a wire loop, use Faraday's law to find the induced
voltage and set that voltage equal to E x circumference. This is an
example problem in many texts. My question is about the direction of
the induced electric field (hard to explain without a diagram)

Agreed, it is hard to form the correct mental picture,
and even harder to make a diagram of it. But such
diagrams are possible; see below.

presumably it is in the direction of the induced current. But if you
were to move the wire loop, say one diameter to the right, it would
still be in the same uniformly increasing uniform field. So if the
current was clockwise (say) before, it would still be clockwise. But
now the electric field direction at the leftmost point is in the
opposite direction of what it was when that location was the
rightmost point of the as-of-yet-untranslated wire loop. But that
means the direction of the induced electric field caused by a
uniformly increasing uniform field varies with t he orientation of a
wire loop placed in that field!

1) Talking about currents is a red herring. Any actual
current would perturb the magnetic field. The original
question asked about the induced electric field, which
is the conventional and sensible way to analyze what is
going on in a time-varying magnetic field.

2) The last 99 times I got a question like this, the key
to understanding the situation was to realize that the
field in a betatron is _not the gradient of any potential_.

In particular, considering a plane transverse to the change
in B. If you integrate the E field around any imaginary
loop in this plane, you find a voltage drop, in gross
violation of Kirchhoff's laws. What's more, the direction
of the voltage drop is clockwise for all such loops (or
all counterclockwise, depending on which direction you're
looking).

If we think of these loops as gears, the gears do _not_
mesh.

So at this point I've told you what it _doesn't_ look
like. It would be more constructive to tell you what
it _does_ look like. The answer, with fancy diagrams,
can be found at
http://www.av8n.com/physics/non-grady.htm
especially
http://www.av8n.com/physics/non-grady.htm#fig-betatron

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

Lots of students have deep-seated misconceptions about this.

It takes some time and effort to master the idea of a field
that is _not the gradient of any potential_.
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