Doug007 said:
But how can a reference frame for the charges be constructed because there arn't any? It's a uniform continuous distribution of charge with no discrete points of any form - charge or otherwise - to refer to!
but the "soup" is moving in reference to the "stationary" observer. it matters not that it is discrete little chunks of soup moving past or one continuous glob of it. if the observer pokes his finger into the stream, he'll know whether or not it is moving.
FYI: here is a reposting of a thought experiment you can do to see how this magnetic field comes from the electrostatic field with special relativity taken into consideration. the magnetic force is actually not a different force than the electrostatic force but is a manifestation of it in the context of moving charges in a relativistic reality.
The classical electromagnetic effect is perfectly consistent with the lone
electrostatic effect but with special relativity taken into consideration.
The simplest hypothetical experiment would be two identical parallel
infinite lines of charge (with charge per unit length of \lambda \
and some non-zero mass per unit length of \rho \ separated
by some distance R \. If the lineal mass density is small enough
that gravitational forces can be neglected in comparison to the electrostatic
forces, the static non-relativistic repulsive (outward) acceleration (at the instance
of time that the lines of charge are separated by distance R \)
for each infinite parallel line of charge would be:
a = \frac{F}{m} = \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho}
If the lines of charge are moving together past the observer at some
velocity, v \, the non-relativistic electrostatic force would appear to be
unchanged and that
would be the acceleration an observer traveling along
with the lines of charge would observe.
Now, if special relativity is considered, the in-motion observer's clock
would be ticking at a relative
rate (ticks per unit time or 1/time) of \sqrt{1 - v^2/c^2}
from the point-of-view of the stationary observer because of time dilation. Since
acceleration is proportional to (1/time)
2, the at-rest observer would observe
an acceleration scaled by the square of that rate, or by {1 - v^2/c^2} \,
compared to what the moving observer sees. Then the observed outward
acceleration of the two infinite lines as viewed by the stationary observer would be:
a = \left(1 - v^2 / c^2 \right) \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho}
or
a = \frac{F}{m} = \frac{ \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} - \frac{v^2}{c^2} \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} }{\rho} = \frac{ F_e - F_m }{\rho}
The first term in the numerator, F_e \, is the electrostatic force (per unit length) outward and is
reduced by the second term, F_m \, which with a little manipulation, can be shown
to be the classical magnetic force between two lines of charge (or conductors).
The electric current, i_0 \, in each conductor is
i_0 = v \lambda \
and \frac{1}{\epsilon_0 c^2} is the magnetic permeability
\mu_0 = \frac{1}{\epsilon_0 c^2}
because c^2 = \frac{1}{ \mu_0 \epsilon_0 }
so you get for the 2
nd force term:
F_m = \frac{v^2}{c^2} \frac{1}{4 \pi \epsilon_0} \frac{2 \lambda^2}{R} = \frac{\mu_0}{4 \pi} \frac{2 i_0^2}{R}
which is precisely what the classical E&M textbooks say is the magnetic force (per unit length)
between two parallel conductors, separated by R \, with identical current i_0 \.
