Re: motional emf

[John Denker <jsd@AV8N.COM>]

Carl E. Mungan wrote:
> Some textbooks further confuse matters by claiming that "a changing
> magnetic flux produces an electric field." (For example, that's the
> title of Sec. 29.7 of Giancoli, Physics for Scientists & Engineers,
> 3rd ed.) This statement would be correct if it said "magnetic field"
> not "magnetic flux".

I agree that the quoted version is problematic.  Note
however that there are *two* ways to fix it:
 a) a changing magnetic field produces an electric field.
 b) a changing magnetic flux (across a given area)
    produces a voltage (around the perimeter of that area).

You should not mix&match;  you should not equate changing
flux (version b) with electric field (version a) as the
Giancoli quote does.

> The erroneous chain of logic is: -time derivative of magnetic flux =
> emf = line integral of electric field

Let's be clear about what's erroneous and what's not.
The words "line integral of" (which were missing in
the first quoted version) are present here, so this
version _if properly interpreted_ looks like a Maxwell
equation to me.  (Note that there are two valid ways
of formulating the Maxwell equations, the differential
form and the integral form.)

If there is an error, it comes from misapplying this
Maxwell equation.  The equation properly applies to
changing the flux through a given loop.  In contrast
if you try to apply it to changing the loop around a
given flux, all bets are off.  Counterexamples abound.
See Feynman volume II page 17-3 for pictures and
discussion.  See below for more on this.

The fact that an equation can be misapplied does not
make the equation invalid.  Heaven knows students
misapply F=ma all the time, but we still consider it
a valid equation.

> There is no induced (nonconservative) electric field in the case of
> motional emf (in the lab frame). (Of course, if the moving bar does
> not contact the slide rail circuit, then charges will pile up at the
>  two ends of the bar and produce an *opposing and conservative*
> electric field, but that's not what's being referred to above.)

Are we getting tangled up in the terminology?  I have
never figured out what "EMF" is supposed to mean.  I
prefer discussing things in terms of "voltage".
  *) We agree that there is no _line integral_ of the
electric field in the scenario we are discussing.
  *) OTOH the moving rail does produce a voltage.  (Many
authors call this voltage an EMF.)

If somebody wishes to redefine "EMF" to refer only to
the nonconservative part of the voltage, I suppose that's
allowed ... but they ought to clearly warn the readers
or there will be massive confusion.

You can have perfectly fine voltages without having a
nonzero line-integral.  Indeed in any situation where
you expect Kirchhoff's "laws" to apply, the line-integral
will be zero ... but there are still voltages.

===========

Actually I'm not particularly enamored of Feynman's
discussion of the "flux linkage rule".  He derives
"rule" and observes that it is something of a chimera,
resting on different foundations in different cases.
And he shows that it is subject to "exceptions".  My
strong preference is to forget about this flaky "rule"
entirely.  That is:
 -- Maxwell equations: good.
 -- Lorentz force law:: good.
 -- Combining these into a "flux linkage rule": bad.