On Fri, 1 Aug 2008, Salmon Egg wrote:
> After all these these years dealing with electromagnetism, I still do
> not fully understand a Faraday disk based upon Maxwell's equations.
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> Something has to be added to the Maxwell equations.
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> This electric field may be called the Lorentz force or something
> similar, but it seems to be something that must be added to the
> equations I have already listed.
Sure, the Maxwell equations alone are not sufficient. They tell you how
the fields depend on the sources. How do the sources depend on the fields?
For that, you need the consitutive relations D = epsilon E, B = mu H,
J = conductivity E.
> This arises simply out of relativity
> where the EM field has to be a tensor and has to transform properly with
> a Lorentz transformation.
.... or how epsilon and mu transform. Epsilon and mu contain the effects
of
electric polarisation and magnetisation. The question concerns a rotating
magnetised disk.
> Questions:
>
> 1. Can this Lorentz force be derived without invoking relativity or
> another law of nature? That is, are Maxwell's equations sufficient?
A qualified "no". In the sense that the Lorentz force provides an
operational definition of E and B, the Maxwell equations don't make any
sense without it, so it's already there. In the sense that the Maxwell
equations don't tell you anything about how fields act on the sources,
it's an extra add-on.
Maxwell included the effects of fields on sources in _his_ set of
equations. The 4-equation Hertz-Heaviside distillation is what omits them
in the usual telling of the story. Neither Hertz nor Heaviside ignored
them though, but considering that the constitutive relations and Lorentz
force are not PDEs, why should they be lumped in with the 4 PDEs that tell
us about the behaviour of the fields?
> 2. Given Maxwell's equations and F = q E, can the forces between wires
> carrying current be derived?
No. Given these and special relativity, yes (and I've seen students
discover that it's harder than they first thought, but educational). You
don't need the entire Maxwell equations, all you need is Coulomb's law +
SR.
--
Timo Nieminen - Home page: http://www.physics.uq.edu.au/people/nieminen/
E-prints: http://eprint.uq.edu.au/view/person/Nieminen,_Timo_A..html
Shrine to Spirits: http://www.users.bigpond.com/timo_nieminen/spirits.html
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