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The
derivation of Purcell however is somewhat dubious. Mainly so
because the velocity of the test charge is unrealistically
taken to be always the same as the charge velocity in the wire.
So, when the test
charge doubles it speed, the current I
through the wire is also doubled and the magnetic force
is quadrupled. This however, makes it impossible to determine if
the contributions to the
higher magnetic force come from either the higher current I,
the higher speed v,
or both.
The
electrons in a real live wire drift with a wide range of
different velocities which together produce
the current I.
We�ll discuss our derivation, which starts of with just the
current I
through the
wire and the speed v
of the test-charge. Surprisingly, this derivation turns out to
be even
simpler as Purcell�s. (for the case of the charge moving
parallel to the wire).
We�ll
also derive the case where the charge is moving perpendicular to
the wire. The required charge
density is derived for a current carrying wire in order to be
neutral in the rest-frame. To be
self consistent we will derive the relativistic EM Potential and
the relativistic Electrostatic Field
for a point particle from the classical EM wave equations in a
way which is both short
and
simple.
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