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Re: mag field of infinite wire (summary)



The problem was to show for an infinite wire along the z-axis that B_z = 0.

Solution #1 - Bowen and Denker's solution translated into integral form:

If B_z exists, it could at most be a function of r. Now draw a
rectangular loop parallel to the wire with one side at r1 and another
side at r2. Apply Ampere's law to this loop. We get B_z(r1) - B_z(r2)
= 0 since the loop encloses no current. Hence B_z must be a constant.
This is exactly the same argument that many textbooks use to prove
that B outside an ideal solenoid must be zero.

Now, how can you show that the constant must be zero? In general, you
cannot. For example, the wire might be running down the axis of a
solenoid. This is no different than the problem we discussed on the
list a short while back, concerning how to show using Gauss' law that
the electric field outside a parallel-plate capacitor must be zero.
You cannot exclude an external field arising, for example, from a
second capacitor inside of which the first capacitor is located.

However, if there are no other sources anywhere, we cannot have a
constant B_z everywhere in space.

Solution #2 - adapted from a private email from Ben Crowell

Consider a charged particle located along the y-axis and suppose it
is moving radially toward the wire. If there is a B_z, then the
particle experiences a nonzero force in the azimuthal direction. But
this is impossible by symmetry, since the wire is symmetric with
respect to reflection across the yz-plane.

You might object as follows. Suppose instead of the wire, there was a
permanent C-shaped magnet with N pole at z = -1 m and S pole at z =
+1 m. This *does* create a nonzero B_z and hence a nonzero F_phi. But
isn't this arrangement also symmetric with respect to reflection
across the yz-plane? The answer is no, it is not. Think about it.
--
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/