Re: Magnetic Field of Finite Wire
- From: Thomas Smid <thomas.smid@xxxxxxxxx>
- Date: Wed, 14 Dec 2005 05:33:59 +0000 (UTC)
Igor Khavkine wrote:
> Thomas Smid wrote:
> > In standard textbooks (e.g. Berkeley Physics Course, Feynman) the
> > magnetic field of a wire is derived from the principles of relativity
> > by considering the length contraction for the charge distribution in
> > different reference frames. This yields a result exactly identical to
> > the one predicted by electrodynamics for the field. However, the
> > derivation is always only done for an infinite wire. For a finite wire,
> > one finds in fact that the results differ:
>
> Such a calculation would make no sense for a finite wire. A finite
> length wire cannot carry a uniform current, which is essential for the
> application of a Lorentz transformation. The current distribution in a
> wire cannot be arbitrary, it must satisfy the charge continuity
> equation.
I don't see why charge continuity would be violated for a finite wire.
If you connect the electrodes of a battery with a length of wire, then
the amount of charges entering the wire will be exactly equal to the
amount of charges leaving the wire i.e. charge continuity is preserved.
In fact, if you assume a steady state, then charge continuity (i.e. the
uniformity of the current) holds by definition.
>
> [...]
> > So I wonder what the correct
> > behaviour in the far field of a current system actually is. Does
> > anybody know some observational or experimental evidence in this
> > respect?
>
> There are things in our every day lives whose functionality depends on
> how well Maxwell's equations model reality. Look at a TV, a radio, or a
> cell phone, all have finite length wires that either emit or absorb EM
> radiation. The reception and transmission of EM signals by these
> devices (that's the experimental part) is modeled by antenna theory
> (that's the theoretical part), which is in turn based on Maxwell's
> equations. If there were any deviation from expected behavior, we would
> have heard of it by now.
First of all, in order to avoid any misunderstanding: we are not
considering the radiation field of an AC current here, but the static
field of a DC current, and it is for this that I would be interested in
the experimental evidence for the far-field behaviour.
Of course, all wires must be finite in reality, but that's exactly the
point I am making here:
the usual derivations I referred to above assume the wire to be
*literally* infinite. In the Berkeley Physics Course, there is even a
footnote in this respect, which argues that the apparent violation of
charge conservation due to the length contraction would not be an issue
here as one can't enclose the total charge of an infinite wire with
some boundary. This argument is of course both mathematically and
physically wrong: if you have a finite system of charges and enclose
it with some boundary, and if you then multiply the size of both by the
same factor which you let go to infinity, then even an infinite charge
distribution will still be enclosed i.e. charge conservation must
apply.
Fo a finite wire, the error by not conserving the total charge may be
small for the near field (i.e. for distances small compared to its
length) but become crucial for the far field as the leading monopole
term would incorrectly be non-zero.
Thomas
.
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- Magnetic Field of Finite Wire
- From: Thomas Smid
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- From: Igor Khavkine
- Magnetic Field of Finite Wire
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