Re: EM Induction - A conceptual question

["Carl E. Mungan" <mungan@USNA.EDU>]

John D wrote:

> There is a widespread misconception that the uniform
> field extends all the way to the ends, so that the flux
> lines shoot straight out the end like water shooting out
> of a garden hose.  This is not right.
>
> You can easily show (by symmetry!) that the field right
> at the end of long solenoid is half the strength of the
> field at a typical point not near the end.  That means
> that half the field lines have "leaked out" sideways
> before reaching the end.  There's a whole lot of fringing
> going on.

But it's interesting to pursue the thought a little further and ask
where the lines are leaking out. An ideal solenoid is wound so
tightly that we can think of the current as a sheet current that is
infinitely wide. Let's drop only the second assumption, ie. suppose
that the solenoid is wound infinitely tightly but only has a finite
length. (A finite sheet current wrapped into a tube.) This problem is
solved in Tipler (on the axis) by integrating the field for a set of
loops, each of which carries a current of n*I*dz. The result is given
in Eq. 29.8 on page 891 (4th ed, Vol 2). The answer depends on R and
the distance to the two ends.

For example, take the radius of the solenoid to be say 10% of its
length. Now how close to one end do you have to be in order for the
field to be 10% smaller than its midpoint field strength? I get an
answer of approximately 4.4% of the solenoid's length.

IOW, most of the "leakage" and "fringing divergence" occurs near the
ends for a tightly wound solenoid. So I'm not sure it's all that bad
to think of it as a hose. I admit the above only treats the field
on-axis, so I'm willing to be persuaded that the situation is worse
off-axis. But I'm having a hard time convincing myself. Has the
problem of a finite, uniform sheet current rolled up been solved and
the field lines off the axis graphed? Carl
--
Carl E. Mungan, Asst. Prof. of Physics  410-293-6680 (O) -3729 (F)
      U.S. Naval Academy, Stop 9C, Annapolis, MD 21402-5026
mungan@usna.edu    http://physics.usna.edu/physics/faculty/mungan/