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Re: motional emf



Kenny Stephens wrote:
This is my first posting to the list so please be tolerant.

I'm currently reviewing some chapters for a "well-known" cal-based
intro physics book. It uses a popular explanation (I've seen it
elsewhere) for deriving the emf developed by a conductor moving with
uniform velocity through a constant magnetic field. Jsut to be
thorough, here's the setup:

Two conducting rods are placed parallel to each other. Let's call
them rails and say the left end of both rails are connected by a
resistance, R. Another conducting rod of length L is placed across
(perpendicular) to the rods and can slide freely along the rails. An
external agent acts on the rod to give it a uniform velocity, v,
parallel to the rails (and away from the resistance). A uniform
magnetic field is applied perpendicular to the plane of the problem
(let's say into the page). Assume a current flows through the circuit
(through R, along one rail, up the rod and returns along the other
rail) such that the charge carries have a drift velocity v_d.

The text says that each charge carrier, q, in the rod has the
velocity v and since q moves in a magnetic field it experiences a
lorentz force F_M= qv cross B. The text then states that the work
done by this force pushing the charges along the rod is F_M * L=
qvBL. Since emf is energy per charge, the motional emf between the
ends of the rod is E= vBL.

Now this bugs the heck out me because magnetic forces are not
supposed to do work.

Imagine a bug riding along with the wire. The bug sees the wire as being
at rest with the magnetic field moving with speed v. Since the bug sees
the wire, and hence the charge (except for the small drift velocity), at
rest in the his frame, it would say, if it could, that there was no
magnetic force on the mobile charge in the wire (presumably free
electrons). Still, one might imagine a voltmeter in the bug's frame of
reference across the wire on which it is riding. One would expect the
reading to be the same as that of a voltmeter connected across the fixed
rails. Since the bug sees no magnetic force, it would attribute the emf
to an electric field E' of magnitude vB' so that the magnitude of the
electric force on the charge in the wire is F'=qvB' and the work done by
this force is qvB'L. The emf would be vB'l. [The primes refer to the
bug's frame of reference. I have used "E" for the electric field, not emf.]

The above is in agreement with Einstein's special theory of relativity
in the case that beta=v/c is small -- in which case
gamma=1/sqrt(1-beta^2) is approximately equal to one -- in other words
the case where special relativity reduces to Galilean relativity. In
such a limit, B' would be the same as B, so the bug would see emf=vBL
as claimed. I have done all this in more detail in a series of postings
on PHYSHARE starting 3/24/04, which one may find in the archives. A
couple of corrections or changes in points of view were made as I went
along. There are references to works by Feynman, Purcell, Nathaniel
Frank, and Elisha Huggins there. Frank points out that the usual
approach to motional emf based on the magnetic force is approximate,
requiring special relativity for an exact treatment. I think the Huggins
approach based on Galilean relativity is equivalently inexact, but
perfectly adequate for ordinary applications. Purcell and others give
the special relativistic transformation equations for the
electromagnetic field. These, as noted, reduce to the Galilean result
for small velocities.

You mentioned that the charge experienced a Lorentz force. Remember that
there are two terms in the Lorentz force:

_F_=q*_E_+ q*_v_x_B_.

The bug sees only the first term, while an observer at rest relative to
the magnetic field sees only the second term. Since they have the same
effect (when v/c is small), one can say, to a good approximation, that
what an observer in the frame of the magnetic field sees as a magnetic
field, the bug on the wire sees as an electric field. I like this
approach (as seen by the bug), because there is no problem about an
electric field doing work on a charged particle.

One of the references I looked at said that in the case of the magnet
frame of reference, the work that increases the energy of the charge was
provided by the external agent that moved the wire. I would think the
same is true in the bug's frame of reference except that he would regard
the external agent as doing such work in moving the magnet.

Frank, not having reached special relativity (in the Galilean limit),
regarded _v_x_B_ as an effective electric field in the case of motional
induction. If I remember correctly, he said this was the preferred way
of treating motional emf.

It also bothered me that the work done in increasing the energy of the
charge might be done by a magnetic force along the moving wire
(perpendicular to the wire's velocity). A magnetic force does not do
work on a free charged particle in a magnetic field, because the force
is perpendicular to the velocity. Although I have not seen it discussed,
from the frame of reference of the magnet, the mobile charge in the wire
is moving along a path in a path with components of velocity in the
direction of motion of the wire as well as along the wire (the drift
velocity)-- in other words in a path not confined to the wire. I wonder
if this has any bearing on the question of what does work on the charge.
I am aware that, in the case of a closed circuit, the induced current
will produce a magnetic field and a back force such as to oppose the the
applied force in accordance with Lenz's law. I can see how a constant
applied force is needed to have uniform motion, this force doing work on
the moving wire. Somehow, this work contributes to the energy acquired
by the charge (and hence the emf). However, the exact mechanism by which
energy is acquired by the free electrons is not so clear when viewed
from the magnet frame of reference. Any thoughts?

Using this explanation just sets the students up
for confusion and puts me in a pickle to try to justify it.

In all honesty, I see the point if one tries to describe the situation
from the magnet frame of reference. But the bug on the wire doesn't have
any trouble seeing that the work is done by the electric field.

I prefer the explanation of calculating the changing flux, Phi_M=
BLx, through the circuit where x is the position of the rod measured
along the rails from the resistance. This gives the emf E=
-dPhi_M/dt= BL(dx/dt)= BLv.

This might do, and it is done in a number of texts, but I seem to recall
that it isn't completely applicable to all cases. I don't have time to
look it up right now, but the Feynman _Lectures_ and _Introduction to
Electricity and Optics_2nd ed. by N.Frank have comments on this. As I
recall, Feynman regards motional emf and induction by virtue only of a
change of flux (as in a transformer) as being separate phenomena that
somehow can _usually_ be subsumed under the same law -- Faraday's flux law.

Hugh Logan