Re: induced emf and eddy currents

[John Mallinckrodt <ajmallinckro@CSUPomona.Edu>]

On Tue, 17 Jun 1997, Carl E. Mungan wrote:

> (1) How can there still be an emf in the loop if it's not closed?
> Theoretically, I don't know how to calculate the flux and use Faraday's Law
> in its integral form because there is no closed loop. It doesn't even make
> sense to me: what if I made two cuts in the loop - would I still get an
> emf? How about if I made a really big cut? Experimentally it doesn't make
> sense either - I would make different measurements with a voltmeter
> depending on how I connected the leads to the loop (eg. if they snaked
> around backwards around the outside of the loop), so how can one speak of a
> definite emf as though it were a battery?

The direct answer to your question is that there *cannot* be an 
emf "in", "around", "through", (or any other inappropriate 
preposition) a nonclosed loop.  An emf is, by definition, the 
integral of the tangential component of the electric field around 
a *closed* (but not necessarily physical) loop.  On the other 
hand, if the gap (or gaps) in a physical loop is (are) small, one 
might be at least partially forgiven for implicitly completing a 
loop with the shortest path(s) across the gap(s).  And of course, 
you are right that your voltmeter will read different values 
depending on the configuration of its leads.  (See "What do 
'voltmeters' measure?: Faraday's law in a multiply connected 
region," Robert H. Romer, Am. J. Phys., Vol. 50, No. 12, December 
1982, pages 1089-1093 for a nice discussion of precisely this 
problem.)

> (2) A different issue. Let's suppose the loop is made of real wire and
> hence is actually three-dimensional. I can then draw closed loops all over
> the inside of the wire. So won't there be gazillions of eddy currents on
> all length scales inside the wire? Won't each individual valence electron
> be driven every which way at once? Exactly what will happen? I can't
> visualize it. Will the internal currents cancel, leaving net currents
> flowing like a skin near the surfaces of the wire? I don't recall having
> heard of that, so I think there's something basically wrong with my
> reasoning here.

The electrons are always driven by the *local* electric field.  
The electric field is determined not only by the spatial 
configuration of the time changing magnetic flux, but also by the 
spatial configuration of static charges.  

Imagine a situation in which there are conductors but, initially, 
no static charges anywhere and no time dependent magnetic fields 
(and, thus, no local electric fields or currents.)  Now start 
changing the magnetic field.  This will produce an unambiguous 
electric field at every point in space (which may, however, be 
difficult to determine unless there is sufficient symmetry.)  The 
electric field will begin to move the electrons and, because their 
motion is likely not to be divergence free, static charges will 
begin to appear.  If the magnetic field changes *steadily* a 
dynamic equilibrium will be reached very quickly.  This is why we 
almost instantaneously get continuous currents around *any* closed 
conducting loop regardless of whether or not it is placed 
"symmetrically" in a region of changing magnetic flux.

> (3) A more basic issue. What exactly is an "emf"? Given that, "The
> (nonconservative) electric fields produced by induction cannot be described
> by an electric potential" (bottom of p. 791), what is emf, if not a
> potential?

I don't know how to answer this other than to reiterate the 
definition of an emf.  Because emf's are not fields--they are 
integrals around closed paths *not* functions of spatial position--
they certainly cannot be potentials.

John
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