Re: [Phys-l] induced electric field

[John Mallinckrodt <ajm@csupomona.edu>]

I knew we could lure David into providing an analytical result if we  
stayed at it long enough!  I have placed a graphic view of the first  
quadrant field lines (for L = 2) at

   http://dl.dropbox.com/u/409806/Squarish.pdf

The graphic includes a shaded region indicating the region of nonzero  
dB/dt and circular lines (in red) for comparison.

I certainly agree with Carl that this is a very nice result that  
would make a most useful "Note" for AJP.

John Mallinckrodt
Cal Poly Pomona

On Nov 24, 2009, at 9:45 AM, Carl Mungan wrote:

> David Bowman:
>
> Fantastic, I thought this thread was dying and then you guys came
> along and resurrected it with some incredible insights.
>
> Okay, can you please provide some details on how you obtained your
> analytic result?
>
> 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. Here
> R-vector is defined as the difference between the field position
> (r-vector) and the source position (r'-vector), and the integral is a
> volume integral over all source elements. Also c is the (constant)
> value of dB/dt where k-hat is the unit vector along the z-axis.
> Cylindrical coordinates are used, with axes aligned along the
> symmetry axes of the square solenoid.
>
> Am I on the right track?
>
> I checked the above procedure for a circular solenoid and it gives
> the familiar solution.
>
> I also noticed (yes, I'm a bit slow compared to you guys) that if I
> take the curl of the familiar solution (namely r*c/2 inside and
> R^2*c/2r outside, both in the azimuthal direction) that I correctly
> get c inside and zero outside. (That would make a good homework
> problem.) Great, I assume that I could similarly take the curl of
> your solution for the square solenoid and get exactly the same
> answers.
>
> David, could I STRONGLY urge you to write up your solution and send
> it to AJP? I'm confident that John M would agree with me that this is
> a very interesting and instructive result that MANY people will be
> interested in.
>
> Much thanks also to Bob S for the fantastic MathCad rendering and
> Viewer program. It really helped to see the field vectors.
>
> *This* is the kind of thread that makes PHYS-L so great. I wish
> sometimes we had more such threads.... -Carl
> --
> Carl E Mungan, Assoc Prof of Physics  410-293-6680 (O) -3729 (F)
> Naval Academy Stop 9c, 572C Holloway Rd, Annapolis MD 21402-1363
> mailto:mungan@usna.edu     http://usna.edu/Users/physics/mungan/
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