It is tempting to speculate that by symmetry the "emf" (a term not allowed
by some PHYS-L participants) induced in a circular arc of central angle A
should be just A/(2 pi) times that induced around the corresponding complete
circle. By the same reasoning, the emf induced in a straight segment would
be just one-fourth of that induced around a square of side equal to the
length of the segment. Actually this seems to give a reasonable result when
applied to a square inscribed in a circle. The difference between the emf's
in a side of the square and the adjacent circular arc agrees with the emf
predicted by the "flux rule" applied to the segment of the circle formed by
this arc and chord. (Or is this just a coincidence for this particular
geometry?) I've been trying to reconcile the result for the arc with that
for the straight segment by examining the limit as the radius of the circle
gets very large or the central angle of the arc gets very small, but have
been unable to arrive at anything useful.
-----Original Message-----
From: Wolfgang Rueckner [SMTP:rueckner@FAS.HARVARD.EDU]
Sent: Wednesday, May 30, 2001 7:52 AM
To: PHYS-L@lists.nau.edu
Subject: Faraday induction
A student asked me a question that I couldn't satisfactorily answer
and
could use your help. It concerns Faraday induction. Suppose we
have a
single loop of wire lying in the plane of the page (or monitor
screen) and
the loop is split -- that is to say, it's not a complete circuit.
Also
imagine an increasing magnetic field into the page (monitor).
Faraday's/Lenz's law tells us that an emf will be induced such as to
produce a CCW current which generates a magnetic field out of the
page. If
you envision the split in the loop being at the top of the page,
then the
end of the loop to the left of the split would be at a positive
potential
w.r.t. the other end.
Now here's the question. How large can this "split" become? For
example,
suppose we open up the split so that it's as large as the diameter
of the
loop -- what area does one use to calculate the magnetic flux? Does
one
just imagine a "short" between the two open ends of the loop so that
one
has a quasi enclosed area? Or suppose it's opened up even more so
that the
loop becomes just a curved wire? At what point does the loop of
wire no
longer enclose an area so that the problem can't be solved in this
manner?
I look forward to your insights. Thanks, Wolfgang