Re: GAUSS LAW

["John S. Denker" <jsd@MONMOUTH.COM>]

At 04:41 PM 1/30/01 -0500, Ludwik Kowalski wrote:
> I would like to share an elementary introduction to Gauss law
> shown below.... Comments, if any, will be appreciated.

One possibly constructive suggestion:  There seemed to be a slight "leap"
or two in the argument at the place where it passed from talking about
lines going through a piece of surface to E-field dotted with the normal
vector.  Some students might get lost at this point.

To address this, a few pictures might be worth a few thousand words.  We
start from the primary idea (lines penetrating the surface) and derive the
secondary idea (line density dot dS).  In my mind's eye I see several
pictures, including:
 -- small line density, short E vector
 -- larger line density, longer E vector
(the point here is that it is a bit of a conceptual and notational leap to
go from lines per unit area to magnitude of E-vector;  in one case the
length of the lines doesn't matter, while in the other case the length of
the vector does matter)

Other pictures include:
 -- lines perpendicular to area, small area
 -- same lines, larger area
 -- same lines, same magnitude of area, but
    area is tilted to so lines are almost in the plane of the area
(the point here is that the orientation of the area matters)

 ===========

While we're on the topic, I'm not 100% convinced that it helps to refer to
dS as a vector.  In some sense, the element of area is a cross product, dx
cross dy.  In D=3 this is a pseudovector.  In D=4 it is not.  The quantity
of interest here is the triple scalar product E dot (dx cross dy).  It is
of course equal to (E cross dx) dot dy and various other re-arrangements...
But (!) any of those are inelegant compared to writing it simply as TSP(E,
dx, dy).  The triple scalar product is well behaved in D=4 as well as D=3.
It has nice geometric interpretations, including:
 -- lines through the the surface, and/or
 -- volume of the parallelepiped spanned by E, dx, and dy.

On esthetic grounds I like quantities like the TSP which have a geometric
meaning.  I like it when I can say intuitive geometric things about
vectors, independent of their components.

Another related issue is manifested by the fact that it is a _double_
integral over dS.  If I am forced to write something involving S I would
prefer to write ddS to indicate that there are two differentials involved,
and the element of area is a second-order small quantity.