Re: [Phys-l] induced electric field

[John Denker <jsd@av8n.com>]

On 11/24/2009 10:45 AM, Carl Mungan wrote:

> I think this is how one could proceed, but please correct me: We can 
> obtain E-vector as 1/4*pi multiplied by a "Helmholtz" (in this case a 
> "Biot-Savart") type integral of R-vector cross-producted with (c 
> k-hat) divided by the cube of the magnitude of R-vector.

Close.

Since this is a two-dimensional problem, you need
magnitude _squared_ in the denominator of the
integrand.

I call this the two-dimensional Biot-Savart integral
or equivalently the flatland Biot-Savart integral.
It's the same idea as the usual Biot-Savart, although
(a) we are in two dimensions instead of three, and
(b) in this case the integrand tells us the E-field 
associated with one element of d(flux)/d(t).  The 
magnitude of this falls off like 1/R, as you know
from experience with the external field of a long
thin solenoid ... which is a check on what I said 
in the previous paragraph.

If you're in a hurry and/or you're not a calculus
whiz, Macsyma will carry out the integrations for
you.

  Bizarre note:  In the free version, called Maxima,
  it miserably fails to do the definite integrals.
  It goes off into a loop using 100% CPU forever.
  However ... it instantly evaluates the indefinite
  integrals!  How strange is that?  Just evaluate
  the indefinite integral at +L/2 and -L/2 and
  you're all set.

Also FWIW:  If I never see another cross product
that will be fine with me.  You can solve this
problem in flatland (where there are no cross
products) just fine.  The E-field in the plane
can be calculated as a function of the flux
-- or d(flux)/d(t) -- in the plane.  In the
numerator of integrand you need a Rot(90) operator 
that rotates the R vector 90 degrees in the
plane.

I get tired of "famous experts" telling me there
is "no" electrodynamics in flatland, because you
"need" a cross product.  I say electrodynamics 
works fine in flatland, and you do not need a 
cross product.  Not ever.  Really not.  The is 
all easy and elegant in terms of Clifford algebra, 
but you can get by without even that.

	Help stamp out cross products.