Re: motional emf

[Hugh Logan <HloganPHY@CFL.RR.COM>]

Carl E. Mungan wrote:

> One last comment on motional emf. 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.)

The same error is found in the title of Sec. 21-4 on p. 628 of Giancoli,
_Physics, 5th ed.

 This statement
> would be correct if it said "magnetic field" not "magnetic flux".
Giancoli does say this correctly in the last sentence of Sec. 21-4
before the example, but not until the error has been repeated as "a
changing magnetic flux produces an electric field" just above the last
sentence.

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



> There is no induced (nonconservative) electric field in the case of
> motional emf (in the lab frame).

As far as I can tell, Giancoli does not make this error in the previous
Section (21-3), "EMF Induced in a Moving Conductor." He regards arriving
at the emf from the moving rod from F=qvB, W=Fl=(qvB)l, emf=W/q=Blv, emf
being the idea of work per unit charge, not making any special point of
the nature of the field doing the work in moving the charge. However, I
think the error creeps into the next section (21-4) when he regards
F/q=vB as an "effective electric field." He states that a force on the
electrons in the moving conductor "implies that there is an electric
field in the conductor," writing "E=F/q=(qvB)/q=vB." In the margin, one
finds in blue letters, "Electric field is prduced by a changing magnetic
flux."

I first noticed the expression "equivalent electric field intensity"
used (in the way Giancoli uses "effective electric field") in
_Introduction to Electricity and Optics, 2nd ed._by Nathaniel Frank,
McGraw Hill, 1950. Using my notation, this would be, in the case of
motional emf, _E'_=_F_/q=(q*_v_x_B_)/q=_v_x_B_. The prime on _E_ is
Frank's. Although he does not treat relativity, he is careful to write,
"Treating this force per unit charge as an equivalent electric field
intensity E' from the standpoint of an observer moving with the circuit,
we should expect the magnitude of the induced emf around the circuit to
be given by the formula
emf=circulation(_E'_.d_r_)=circulation[(_v_x_B_).d_r_] ."
[ "Circulation" is my translation of "line integral around a closed
path." Frank writes the integrals in terms of tangential components, as
he does not use a full vector treatment.] He states(p. 148)that the
integrated contributions of (_v_x_B_).d_r_, i.e. the emf induced, as in
the case of the conducting rod moving through a magnetic field, "is
equal to the rate at which the conductor cuts lines of B." Frank is also
careful to point out that his approach is an approximation to the
correct relativistic approach, although "extraordinarily well satisfied"
for most practical situations where v/c is small.

I think the equation "time derivative of magnetic flux = emf = line
integral of electric field" is correct for a rigid loop moving uniformly
with respect to the lab frame in which the magnet is fixed if "electric
field" is replaced by "equivalent electric field" as discussed above,
and the observations are in a frame in which the loop is stationary. By
"line integral" I understand it to mean "circulation" as described
above. My definition of emf of the loop is the circulation of the
non-conservative and conservative fields in the loop, which amounts to
the circulation of the non-conservative field(s), since the circulation
of the conservative field is zero. (I am aware that there is more to
emf, but I think the definition I chose suffices for the present
discussion. In the case of the moving rod, the emf would be the line
integral of the non-conservative field along the rod, but the result
wouldn't differ from the circulation definition if the non-conservative
field were confined to the rod.)

Frank states (p. 149), for a stationary field, "If the circuit moves as
a rigid body, undeformed during its motion, the induced emf as given by
[ emf=circulation(_E'_.d_r_)=circulation[(_v_x_B_).d_r_] ] is just equal
to the rate of change of flux through the circuit. ... If the magnetic
field varies with time, one obtains the induced emf by computing the
total time rate of change of flux linking the circuit." (He doesn't use
Stokes' theorem, because of the math level of the book.) On p. 150, "The
situation for induced emfs in circuits which are deformed and do not
move as rigid bodies is more complicated, and in general it is safest to
employ [ emf=circulation(_E'_.d_r_)=circulation[(_v_x_B_).d_r_] ] [but
not the flux rule.] Only in special cases is it true that the induced
emf is equal to the rate of change of flux through the circuit." He
mentions that the rod on the rails with the circuit completed is an
exception, as has been pointed out. I am not sure that texts on this
topic have improved since 1950.

The text, _Introduction to Electromagnetic Theory_ by George Owen*,
Allyn and Bacon, Boston, 1963 (out of print, but recently reissued by
Dover), starts out with Faraday's law (p. 231)in the form
emf=-(d phi)/dt. He uses the definition of flux and Stokes' theorem to
get it in the form

" {Surface integral over S}((curl _E_)._n_dS=-d/dt={Surface integral
over S}(_B_._n_dS),

where S is a "cap surface" bounded by the path L.
He points out the difficulty of taking the total derivative d/dt in the
case the loop is moving, because the limits of integration are time
dependent. [It would be worse in the case the loop was being deformed, I
presume.] One can't just move d/dt across the integral sign. For the
case where the loop frame moves with velocity _v_ with respect to the
lab (magnet) frame, he solves the problem by transforming to the frame
in which the loop is at rest by the Galilean transformation. (There is
almost no special relativity in this book.) Calling the loop frame the
primed frame, he comes up with _E_'=_E_ + _v_x_B_ using a bit of vector
calculus and identities. (There are a couple typos in my copy - a
missing prime on d_l_ at the bottom of p. 232 and an extra dot in a
vector identity.) In the case that _B_ is time independent, D_B_/dt=0,
_E_=0 so that _E_'=_v_x_B_. I found it easier to take the limiting case
of the special relativity transformation as v/c approaches 0.

Some have complained about the difficulty of the relativistic approach.
The very elementary treatments hardly use the results of special
relativity. They just use the common sense idea that both observers see
the same emf. (I don't think this is exactly true, but it is OK if v/c
is small. Since one observer is at rest relative to the rod, v=0 for
him, so he attributes the force acting on the charge as a electric force
qE'. The lab observer sees it as a magnetic force qvB. The distance L
between the rails is the same for both. The moving observer calculates
the emf as qE'L/q=E'L, and the lab observer gets the emf as qvBL/q=vBL.
Equating the emfs, E'L=vBL or E'=vB. I think the difficulty is more
conceptual than mathematical.


 Hugh Logan

*I studied E&M with Dr. George Owen. Back in 1953, I recall his
favorable comments about the excellent physical discussions in Nathaniel
Frank's _Introduction to Electricity and Optics_. As a matter of fact,
he once used it for his junior level course, expecting students to use
his more advanced vector treatment. I took E&M for credit with Dr. Owen
later, using Reitz and Milford, although his notes became the book
referred to. I just looked at the treatment of Faraday's law in the more
mathematical text, _Electromagnetism_ by Slater and Frank, but it seemed
superficial compared with _Introduction to Electricity and Optics._