Re: Pedagogy

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

Fernanda Foertter [Advanced Physics Forums] wrote:
> on a humorous note, if that conversation happens to be over Jackson, then
> all of us are allowed to rant over the difficulty level of the problem ;)
>
> But you are correct...sometimes it is because a student has spent so long
> trying to understand the problem that they are defeated when they finally
> work up the courage to ask the question.
>
> Ever start a problem, that after two pages you just *know* the answer is
> easier than you are making it?  And then someone shows you the answer and
> you can't help but feel dumb about how you went about it?
>
> We've all been there...that one problem that you just can't see the answer
> to it and you are into your third hour of research about it...so when a
> student comes up to a prof that defeated, the last thing they need to hear
> is "oh, thats easy!"


Also on a humorous note, I recall the time a brilliant professor was
done in by one of Jackson's problems, namely problem 1.10 on p. 25 of
the first edition of his _Classical Electrodynamics_. The professor was
at his best when famous people --like Kip Thorne, Edward Teller, and
Eugen Merzbacher -- came to the university to give colloquia,
asking profound questions for a long time after the lecture. Sometimes,
he gave extremely brilliant lectures in class, but on other occasions he
came to class unprepared, relying on his brilliance to get him through.
No one, including some very bright students, could do the problem: "Use
Gauss's theorem to prove that at the surface of a curved charged
conductor the normal derivative of the electric field is given by

(1/E)DE/Dn=-(1/R(1)+1/R(2)) where R(1) and R(2) are the principal radii
of curvature of the surface." [I have used "D" to denote a partial
derivative and parentheses to enclose subscripts.]

The professor and the brightest students discussed this problem for
about two days in class, even considering definitions of curvature from
tensor analysis or Riemannian geometry. I decided to research this
problem at the library. None of the references on p. 23 in Chap. I were
of any help. However, the Bibliography on p.623 listed a book,
_Electrostatique and Magnetostatique_ by E. Durand, Masson, Paris
(1953). I still have a note in my text, "See Durand, p.85." The problem
was there with the solution. It was simple as long as one knew the
definition of principal radii of curvature. I have forgotten the
details, but I seem to recall that it was little more than taking the
cross product of two vectors tangential to curves along the surface. The
professor thought I was conscientious in finding the solution, but felt
it would have been better if I had done it myself. I wasn't very good at
the more mathematical boundary value problems, but I was pretty good at
the basic physical principles. Problem 1.11 was to prove Green's
reciprocation theorem for a volume charge distribution and a surface
charge distribution together. This is a standard problem in texts on
vector analysis, using Green's vector identity -- such as on pp. 139-140
of _Vector and Tensor Analysis_ by Lass. For some reason, this was not
considered rigorous enough. I was fortunate enough to have studied E&M
(though not much beyond Maxwell's equations)with George Owen, who made
considerable use of matrices. What little I knew of matrices at that
time, I learned from Dr. Owen's _Principles of Scientific Mathematics_,
a little book he wrote for a high school summer program, but useful for
college physics students. He used matrices for several topics in a text
he later wrote, _Introduction to Electromagnetic Theory_, Allyn and
Bacon, Inc., Boston, 1983 (now out of print). He treats Green's
reciprocity theorem for point charges on pp. 141-142 very neatly, using
matrices. There is a comment that it applies also to a system of
conductors, which seemed clear. I decided to see if I could generalize
it to Jackson's more general problem. My professor was impressed, and I
was flattered to see my homework draped from his lectern as he explained
the solution to the class. But these were atypical instances. The course
was supplemented with Vol. 2 of the Feynman lectures, which was less
mathematical. Our professor thought that he detected an error in a fine
point about Gauss's law in that volume. I needed to learn more
mathematics like Bessel functions to do the cylindrical problems in
Jackson. Only the first half of the course was needed for an M.S.
degree, so I never went all the way through David Jackson's book.

Hugh Logan