Re: [Phys-l] induced electric field

[John Mallinckrodt <ajm@csupomona.edu>]

On Nov 22, 2009, at 8:36 PM, John Denker wrote:

> I think the shoe is on the other foot.  Some folks are
> assuming without proof -- indeed without evidence -- that
> the squarish solution exists.
>
> ... proponents of squarish solutions are encouraged
> to exhibit some such thing, in specific and formal terms,
> and show that it solves the relevant Maxwell equation in
> the given situation.  I don't think it exists:  too much
> curl near the corners of the "square" and not enough
> elsewhere.

Well, in fact, I actually did do some numerical calculations based on  
a Biot-Savart-like transformation of Faraday's law and they clearly  
do support the "squarish solution."

It shouldn't surprise anyone that the solution we seek is identical  
in form to that for the B-field produced by a uniform current density  
within a region having a square cross section.  For a quick and dirty  
solution I set up a spreadsheet with a 20x20 array of 400 "wires"  
carrying "current" in the +z direction and used it to calculate the  
resulting field at arbitrary positions in the x-y plane.  Inside the  
square, the solution is a little too too susceptible to the varying  
distance to the nearest wire to be reliable, but outside the solution  
is quite stable and smooth.  For a radius of 15 units, I calculated  
the B-field every 3 degrees from 0 to 45 degrees and found that the  
field is purely tangential at 0 and 45 degrees as expected, but has a  
positive radial component for angles in between (and a negative  
radial component for angles between 45 and 90 degrees.)  In addition,  
I found that the field increases monotonically in magnitude as one  
moves from 0 to 45 degrees. These are exactly the signatures one  
would expect for the squarish solution and I would further expect the  
boundary conditions to require the "squarish solution" to propagate  
into the inner region.

John Mallinckrodt
Cal Poly Pomona