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Re: [Phys-l] violation of N3 ??

Regarding John Denker's point of confusion:

I am very impressed by David Bowman's response to the pseudo-force
question. I've read it a few times, and plan to read it a few
more.

I am confused by this point:

On 10/26/2006 10:52 AM, David Bowman wrote:

3) We already know that N3 is violated anyway by magnetic forces
acting between electrically charged particles/current elements
moving nonrelativistically slowly.

I'm not sure what this refers to. Does it perhaps refer to things
like the machine shown in Feynman volume II figure 17-5 in section
17-4?

That is not quite what I had in mind.

Rather I had in mind the leading order (nonretarded instantaneous)
magnetic forces exerted between a pair of current elements or between
a pair of electric charges in mutual motion, interacting via the
Biot-Savart magnetic induction field and the Lorentz force (BSL).
Suppose we have a filamentary current element I*dr whose current is I
and whose differential increment of length along the direction of the
current flow is dr (dr is a vector here) and which is located at
location r (r is a vector). Also suppose we have another current
element I'*dr' correspondingly located at r'. The BSL magnetic force
dF (dF is a vector here) exerted on the Idr element by the magnetic
field produced by the I'dr' element is:

dF = [mu_0]*I*I'*(dr x (dr' x (r - r'))/(4*[pi]*|r - r'|^3)

where the x's above are vector cross products (sorry for not writing
them as 2-forms/bi-vectors). Likewise the BSL magnetic force dF'
(dF' is a vector here) exerted on the I'*dr' element by the magnetic
field produced by the Idr element is:

dF' = [mu_0]*I'*I*(dr' x (dr x (r' - r))/(4*[pi]*|r' - r|^3) .

These forces clearly do not obey the rule dF = -dF' because of the
idiosyncracies of the nature of vector cross products.

Similarly, suppose we have point charges q & q' located at r & r'
respectively, and moving with velocities v & v' respectively. The
BSL magnetic force F exerted on charge q by the magnetic field
produced by q' is:

F = [mu_0]*q*q'*(v x (v' x (r - r'))/(4*[pi]*|r - r'|^3) .

And the BSL magnetic force F' exerted on charge q' by the magnetic
field produced by q is:

F' = [mu_0]*q'*q*(v' x (v x (r' - r))/(4*[pi]*|r' - r|^3) .

As an example of the moving point charge situation consider a charge
q moving to the right in the diagram below with charge q' moving
upward in front of the path of q. Consider the moment that q' is
directly on the line of the path of q and passing it by, vis:

|
^
|
|
---->-- q --->--- q'
|
|
^
|

This ASCII diagram needs to be viewed with a fixed space font. The
arrows in the diagram indicate the paths and movement directions of
the respective charges. For definiteness, suppose that q & q' are
both positive. At the moment considered here there is at the
location of q a magnetic field generated by q' that is pointing out
of the screen toward the viewer. This causes a force to be exerted
on q that is down toward the bottom of the screen. But at this
same moment of time q', being on the direct path of q, is on the
symmetry axis of the magnetic field produced by q that circulates
around that axis. At all points on that axis the magnetic field
generated by q is zero. Thus q', being in a no-field location,
has no magnetic force exerted on it by q. This is a very strong
violation of N3.

Now it's true that for this latter case the magnitude of these
magnetic forces exerted between q & q' are of the order of

(geometric factors)*(|v|*|v'|/c^2)*(the electrostatic Coulomb force)

acting between these charges. Since this force is down by a factor
of 1/c^2 relative to the Coulomb force between them one may be
tempted chuck it away with the wave of a hand claiming that it is
merely a relativistic effect that ought not enter a discussion of
Newtonian-type forces. But consider that this force is of order
v^1 for each charge rather than of order v^2 as typically occurs in
SR corrections. Also, in the case of filamentary currents in
conducting wires there is an overall charge neutrality that cancels
out the Coulomb forces between the otherwise bare moving charges.
In the current-in-wires case the only action-at-a-distance EM forces
acting between these currents is purely this BSL magnetic force.
(There is also the force of constraint offered by the conductor's
work function keeping the charges in the wires but we won't count
that.) We really can't very well dismiss such magnetic forces as
being a negligible relativistic effect since it is responsible for
making nearly all kinds electric motors work (that along with the
forces of constraint). They are very relevant macroscopic effects
with *very* *non*relativistic drift current speeds.

If so, can't it be understood as a non-violation of N3 as
explained in chapter 27 ("field energy and field momentum")?

Or ..... what am I missing?????????????

Certainly one can easily deal with momentum conservation for
momentum transfers between particles and force-fields when one also
considers the field momentum possessed by the fields themselves.
But there seems to be a problem that develops with an *N3* assignment
saying that a particle exerts a force on a field at some location
because it requires that the field at that location also transmit
that force to its various neighboring spatial neighbors at different
locations in space as the momentum is locally transferred across the
spatial reaches of the field. I don't know how one can talk of a
field value at some location as being accelerated according to
Newton's laws by the forces acting on the field at that point in
space. It seems to me that the momentum flows in the field across
space can not be described by Newton's laws at all, and I can't see
how to make sence of such momentum flows across space in terms of a
bucket brigade of local action-reaction pairs. I think Maxwell's
equations are somehow more appropriate. I have a hard time
visualizing the response of a force field to momentum transfers to it
(from external point particles) in Newtonian terms unless the force
field in question is some sort of continuum elastic stress field of
a material medium possessing a Newtonian particle substrate.

David Bowman