When I first use Gauss' law to derive the E field outside an infinite
uniformly charged rod, I like to take a few minutes to go through
some symmetry arguments to show that E can only point in the radial
direction and can only vary in magnitude with the radial coordinate.
I would like to do the analogous thing for Ampere's law to derive the
B field outside an infinite straight wire carrying a current. It is
easy to argue that B again can only depend on the radial coordinate.
But I am having some trouble with the directional argument. I can
eliminate the radial direction easily enough. Here are two arguments:
1. Reverse the direction of the current. According to Ampere's law,
we require B to consequently change sign. But reversing the current
is equivalent to rotating space by 180 degrees so that the wire flips
end-to-end. This properly reverses the sign of the azimuthal and
longitudinal components of B, but not of the radial component. Hence
there can't be a radial component.
2. According to Gauss' law for magnetism, there can't be a flux
across a cylinder enclosing a finite length of the wire. Hence there
can't be a radial component.
Okay, so your assignment should you choose to accept it:
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.
Is there a solution? Carl
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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/