Correct derivation of magnetism from SR

[Hans de Vries]

The correct derivation of magnetism as a relativistic side effect
of electrostatics:

http://chip-architect.com/physics/Magnetism_from_SR.pdfI've claimed quite a few times here that Purcell's derivation is not
correct. Making claims is one thing. Better is to pinpoint exactly
what is wrong, and then, of course, provide the right derivation.

Two mistakes which cancel plus an omission eventually produce the
required result. We’ll discuss the mistakes and then give the correct
derivation, which, surprisingly (for me as well), turns out to be even
simpler. (for the case of the charge moving parallel to the wire)

Also derived is the case where the charge is moving perpendicular
to the wire. Furthermore, the required charge density is derived, for
the electrons in a current carrying wire, in order to be electrically
neutral in the rest-frame.

To be self consistent, the paper does 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.

I spend a lot of effort to make the paper understandable and clear.Regards, Hans.

P.S: An online presentation of Purcell's derivation is available here:
http://physics.weber.edu/schroeder/mrr/MRRhandout.pdf#search="purcell simplified"
from Dan Schroeder (The one from Peskin & Schroeder)

 

[CarlB]

Allow me to be the first on PF to compliment your paper.

And this paragraph from page 5 got past the syntax checker, corrections are in bold:

We now give the correct derivation for the case where the test-charge moves parallel with the wire. This derivation is actually even simpler than the erroneous one. We’ll see that we can ignore all the second order effects which led to the dependence of the force on the square of the velocity of the test charge. In fact we do not need to deal with the speed of the electrons at all. Just with the current I which the wire is carrying. We start off with the familiar Lorentz transforms:

Carl

 

[Hans de Vries]

Thanks, Carl

You'll find that the corrected spelling is already uploaded. It's 4:30 AM where
I'm currently. Time to sleep :^)


Regards, Hans

 

[pervect]

A few comments: It's a bit circular to use Maxwell's equations and then claim that one has 'derived' the magnetic field. Maxwell's equations already assume the existence of a magnetic field. So the title of the paper doesn't exactly represent what's being done. (I'm not sure I have a fix for this concern, but I don't think the title as-is is quite right, I'm just not sure what would be better offhand).

I would also stop short of claiming that Purcell is wrong, though from my brief glance I would agree that your deriation is better, and that Purcell's derivation would be correct only for very low velocities while your derivation does not suffer from that limitation.

One other comment. We seem to agree that the positive and negative charge densities per unit length must be the same in the rest frame of the wire in order for the wire to be uncharged.

Yet your diagram in figure 3 (left) does not seem to reflect this. Shouldn't it be the same as the diagram in figure 1 (left)? This removes some of the apparent "conflict" with Purcell - the diagrams are not really so dissimilar as all that..

 

[Hans de Vries]

I would also stop short of claiming that Purcell is wrong, though from my brief glance I would agree that your deriation is better, and that Purcell's derivation would be correct only for very low velocities while your derivation does not suffer from that limitation.

With the very low velocities you probably mean slower as the electron
drift in the wire. Where it really goes wrong is the intermediate result:

$$F_{elec} =\frac{v^2\ Q_L}{2\pi\epsilon_o c^2 y}\ Q$$

Here the force is proportional to v² while v is the speed of the
test-charge relative to the wire. The force should of course be
linear in v. The derivation gets rid of one v by combining it with QL
into current I. However, the speed which defines the current in
the wire is another v: The relative speed between the electrons and
the ions...

One other comment. We seem to agree that the positive and negative charge densities per unit length must be the same in the rest frame of the wire in order for the wire to be uncharged.

Yet your diagram in figure 3 (left) does not seem to reflect this. Shouldn't it be the same as the diagram in figure 1 (left)? This removes some of the apparent "conflict" with Purcell - the diagrams are not really so dissimilar as all that..

Ah, this is (was) a copy/paste error. The comment in the image belongs
to another one. A corrected version is already uploaded. Thanks.


Regards, Hans

 

[Hans de Vries]

Here the force is proportional to v² while v is the speed of the
test-charge relative to the wire. The force should of course be
linear in v. The derivation gets rid of one v by combining it with QL
into current I. However, the speed which defines the current in
the wire is another v: The relative speed between the electrons and
the ions...

Hmmm, in Dan Schroeder's presentation of the derivation:

http://physics.weber.edu/schroeder/mrr/MRRhandout.pdf#search="purcell simplified"

The test-charge is always at rest with the positive ions, and thus at rest
with the wire... consequently there is only one sort of v here. So it's
hard to mix them up..Regards, Hans

 

[Hans de Vries]

The simplest, and the full derivation of Magnetism
as a Relativistic side effect of ElectroStatics

http://chip-architect.com/physics/Magnetism_from_SR.pdf​

Some rework is done. We now start directly with the derivation, which is further simplified, and critics on the Purcell derivation are limited to the abstract and more to the point.Regards, Hans.