Re: mag field of infinite wire

["John S. Denker" <jsd@MONMOUTH.COM>]

"Carl E. Mungan" wrote:
> Prove that there can't be a longitudinal component to B, ie. B_z = 0
> if the wire is along the z-axis.
>
> You may use Maxwell's equations in their integral or differential
> form and you may use symmetry arguments. You may not use Biot-Savart
> law nor appeal to experiment.

It can't be proven, because strictly speaking it's not true.
There could be a B_z field.  But it would be a bizarre
coincidence.  Let's prove what we can prove:

The obvious starting point is the Maxwell equations, of
which only one is relevant:
        del cross B = J

Now that is a vector equation.  Using cylindrical coordinates
(r, theta, z) the second component (theta component) of that
equation tells us that
        (del/del z) B_r - (del/del r)B_z = 0
and we know twice over that the first term on the LHS is zero,
so the other term must be zero also.  Therefore B_z is a
constant independent of r.  By symmetry it must be independent
of z and theta.  So it is everywhere a constant.

And most significantly, B_z is independent of J.  You can
imagine whatever universal constant B_z you like, but it's
got nothing to do with the wire.  You can change J independently
of the hypothetical B_z.