Re: [Phys-l] induced electric field

[David Bowman <David_Bowman@georgetowncollege.edu>]

Regarding John M's comment from yesterday:

> 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.
>
> It may very well be that I am simply misunderstanding John Denker's
> points, but I keep reading them as suggesting that the lines will
> indeed be circular and that there is nothing fundamentally wrong
> with the derivations that Carl questioned.
>
> John Mallinckrodt
> Cal Poly Pomona

Well I, too, had been ignoring this thread until today, when having
some spare time (actually wanting to take a break from grading) I
decided to look up what the discussion was all about.

I agree with John M's comments above.  However, instead of attempting
a numerical solution I attempted an exact analytic solution for a
spatially uniform time dependent B-field inside a L X L square cross-
section solenoid of infinite length.  It ends up that the E-field is
exactly integrable in closed form, but as one might expect, the
solution is a mess albeit with the expected square symmetries imposed
on an approximately azimuthal field pattern.

The solution is below.  But to cut down on the mess I'll define
some intermediate quantities.  The solenoid & the B-field is
oriented along the z-direction and its (only) z-component is
increasing at a rate of B'.  The z-axis is the center axis of the
solenoid.  The electric field has x & y components E_x & E_y
respectively.

Let A == L/2 - x
Let B == L/2 + x
Let C == L/2 - y
Let D == L/2 + y

Then

E_x = (B'/(4*[pi]))*(A*ln((A^2 + C^2)/(A^2 + D^2)) +
     B*ln((B^2 + C^2)/(B^2 + D^2))+ 2*C(arctan(A/C) + arctan(B/C))
     - 2*D(arctan(A/D) + arctan(B/D)))

and



E_y = (-B'/(4*[pi]))*(C*ln((A^2 + C^2)/(B^2 + C^2)) +
     D*ln((A^2 + D^2)/(B^2 + D^2))+ 2*A(arctan(C/A) + arctan(D/A))
     - 2*B(arctan(C/B) + arctan(D/B)))

David Bowman