Re: [Phys-l] induced electric field

[John Denker <jsd@av8n.com>]

On 11/22/2009 02:39 PM, John Mallinckrodt wrote:
> I wasn't playing close enough attention as this thread was evolving,  
> but I just took the time to read from the the archives and I am  
> confused and unsettled about what I see, especially with regard to  
> Carl's discussion of the figures at
>
>    http://home.minneapolis.edu/~carlsoro/note.htm
>
> I completely agree with Carl's expressed concerns.  Subject only to  
> what seems to me the completely uncontroversial interpretation that  
> we are asking about the E field that is induced by the specified time  
> changing B field and are disregarding any other preexisting or  
> background fields, it is clearly (is it not?) the case that the lines  
> of the induced E-field are closed and exhibit considerable symmetry,  
> but are non-circular.  Indeed, I would expect them to approach  
> circularity as r -> 0 and as r -> infinity, but to be somewhat  
> "squarish" for radii near L/2.
>
> In any event the calculation of E shown in Note 2 is clearly wrong  
> because it assumes in step 2 a symmetry that does not exist. The same  
> objection applies to Note 3.


I think the shoe is on the other foot.  Some folks are
assuming without proof -- indeed without evidence -- that
the squarish solution exists.

In stark contrast, we need not assume the cylindrical
solution exists.  We know it exists.
  E = x j + y i                 [1]
as previously discussed.  If this is not explicit enough or
specific enough or formal enough, please explain.  We all 
know that equation [1] is not the only solution, but it 
nevertheless remains a valid element of the solution set.

Meanwhile, 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.

Equation [1] agrees with my intuition ... but even if 
it didn't, given the choice between an equation and 
unsubstantiated opinion, I go with the equation every
time.