Can SR fully explain magnetic force?

[Sheyr]

As we know for over 100 years magnetic force in an relativistic effect caused by the movement of charged particles.

Let’s take the standard example - current in a wire. If a charged test particle (say electron) is moving relative to the wire it feels the Lorentz force.

I can understand what happens when electron is moving parallel to the wire – electron feels the force which is perpendicular to the wire because of Lorentz contraction of concentration of electrons and/or ions in the wire.

But I can not understand why is the Lorenz force parallel to the wire if the test particle is moving perpendicularly to the wire and why is the Lorentz force nil if the test particle is moving in the direction that we used to call “direction of the magnetic field lines”.

 

[pervect]

Let's draw a diagram:x...^------>

the x represents the current carrying wire, pointing out of the plane of the paper (downward). Let's call this the "-z" direction.

The carret represents the magnetic field due to the wire. It points upwards (let's call this the 'y' direction).

the arrow ------> represents the velocity of the moving charge. Lets' call this the 'x' direction, and the x-velocity of the charge 'v'.

Then we expect a v x B force pointing "up" out of the paper on the charge (a force in the +z direction).

We can explain this by considering how the current looks in the frame of the moving charge.

In the rest frame of the wire, the wire is represented by a straight line

x=y=0

In the rest frame of the charge, the wire is also represented by a straight line, but the line is "tilted"

Since x is constant, and x = \gamma(x' + (v/c^2)t'), we see that x' is not constant for the wire, but is a function of time.

Thus we have to consider the apperent "tilt" of the wire from the viewpoint of the charge. This explains the additional force component that changes the electrons z-velocity.

 

[Hans de Vries]

I can understand what happens when electron is moving parallel to the wire – electron feels the force which is perpendicular to the wire because of Lorentz contraction of concentration of electrons and/or ions in the wire.

Beware here for the often used but erroneous 'split wire' argument, which
wrongly claims that the electron-part-of-the-wire and Ion-part-of-the-wire
have different Lorenz contractions and thus different charge concentrations.

The wire never splits. It's the shape of the Coulomb fields of the individual
charges which flattens. As long as the test-charge moves parallel this
flattening is in the direction of the wire.

However, if the test-charge moves perpendicular to the wire then the
flattening is under an angle. The net result, subtracting the ion field from
the electron field now gives a force parallel to the wire.


Regards, Hans

 

[Sheyr]

Thank you perfect, thank you Hans. I have still problem to understand the +z component of the force effecting on the charge - i.e. where it comes from...
Do you know any paper or www site where can I see some drawings or detail explanation of the problem?

 

[Hans de Vries]

Thank you perfect, thank you Hans. I have still problem to understand the +z component of the force effecting on the charge - i.e. where it comes from...
Do you know any paper or www site where can I see some drawings or detail explanation of the problem?

Let me try to picture it without the math.
(I have never seen a www site /paper unfortunately)

So, how does SR manage (purely electrostatically) that a wire with a
current exerts a force (with a direction parallel to the wire) on a small
test-current (with a direction perpendicular to the wire)?
1) First let's replace both wires with an electron/ion pair for simplicity.

2) Let's start with the generalized Coloumb potential: There is an effect
both in electrostatics and gravitation that is as follows: If you look at
the place where the sun is and in what direction it pulls us then these
are not the same! We see the sun where it was 8 minutes ago (light
travel time), but we are pulled to the location were the sun would be
if it had continued to travel in a straight line for another 8 minutes.

Why? Well because a moving mass/charge "emits" a force at an angle
which differs from the usual mass/charge at rest. The angle does
always stays the same when the force-field propagates towards you
and its tail will always point to the place where the mass/charge would
be if it continued to move at a straight line.

If the force-field of the sun has traveled (at the speed of light) towards
you for the 8 minutes it takes then it will point to the place where the
sun should be, also 8 minutes later, if it continued to move in the same
direction. Actually the sun moves in a circle so there's a small difference.

3) Now let's go back to the current in our (minimal) wire. The ion in the
wire doesn't move thus the electric force is always toward (away from)
the ion. The electron does move however, so the electric force is always
towards (away from) a point somewhere ahead of the electron.

4) What does happen at our test-wire if there is no test-current (both
test-electron and test-ion at rest) If we subtract the two electric
forces, the force from the ion at rest and the force from the moving
electron which is directed at some point ahead of the electron then
you'll see that the net-result is an electric force PARALLEL to the current.

5) As far as our test-wire is concerned: The force on the test-ion and
test-electron is opposite since they are at rest and thus the net force
on the test-wire is zero.

6) This balance is disturbed if we introduce a test-current. Let's keep
the test-ion at rest and let the electron move perpendicular AWAY from
the wire. You may now want to take paper and pencil to draw the
forces on the moving test-electron from both the ion and the electron
in the wire. The force from the ion stays directed in the same direction
but the force from the electron is now under a SMALLER angle.

7) Next we let the test-electron move perpendicular TOWARDS the wire
again we see the same angle for the ion but now the angle of the force
from the electron is LARGER.

8) The test-current, towards or away from the wire disturbs the balance
we did see if there is no test-current. In (6) there is a force on the
test-wire opposed to the way in which the electron in the main wire is
moving while in (7) the net force is in the same direction as the electron
in the main wire is moving.That's it. I hope it helps. You may google for "Lienard-Wiechert" for the
Generalized Coloumb potential or for a "Speed of Gravity" discussion to
look at a typical misunderstanding of the effect which sometimes make
people claim that Gravity has an infinite speed. (Because it points to the
location where the mass would be if it continues at a straight line)Regards, Hans

 

[pervect]

http://www.phys.ufl.edu/~rfield/PHY2061/images/relativity_14.pdf

describes the electric field of a moving charge. If you do enough work, you can use that numerical relationship to answer the question numerically if you work the problem in terms of the frame in which your test charge is stationary. In that particular frame, all forces on the charge will be electrostatic forces, wo you can, in principle, just add up all the electric forces from the various objects. If you need to see more of the derivation, try earlier and later slides in the thread, i.e. replace the 14 by 13,12,11,10,9, etc.

Actually doing the math is probably a lot of work. You'll also have to properly account for how charge densities transform (but this is described in some of the above references). If you study the field of a moving charge (see also the diagram in #15), you'll see that it is concentrated in a direction transverse to the motion.

Remember that the apparent motion between the test charge and the charge flowing in the wire has two components - one component due to the motion of the charge flowing along the wire, another component due to the relative motion of the wire and the test charge.

"Static" charge due to the wire only has one component (the motion of the test charge relative to the wire).

This is basically Hans' explanation, which I think is better than the one I offered.

I don't have any really great online references offhand for your specific question. (The fla references I quoted earlier are among the simplest I've seen for E&M). You might try a textbook like Purcell's

https://www.amazon.com/gp/product/0070049084/?tag=pfamazon01-20

and if you're not willing to fork out $140 (can't say that I'd blame you), try your local library, it should be reasonably easy to get via interlibrary loan.

 

[pervect]

A general observation:

There are two ways to solve this problem, the easy way, and the hard way(s).

The hard way(s) have already been described - you solve the problem again by scratch from the sources, given Maxwell's equations, in the new coordinate system.

The easy way is this. If you have a solution of the electromagnetic field equations in one coordinate system, you can modify that solution via the equivalent of a "Lorentz boost" to find the solution in another coordinate system.

Thus if you know all the components of E and B in one coordinate system, you do not have to re-solve Maxwell's equations from the source charges and currents to find the solution in a new coordinate system, all you have to do is to take the existing solution and modify it.

The equations by which this are done, i.e. how the E and B fields transform with velocity, are explicitly spelled out in summary form at

http://scienceworld.wolfram.com/physics/ElectromagneticFieldTensor.html

(which, however, really should add that the transformation laws are given for a charge traveling in the z direction at a velocity of v).

The proof of the validity of these transformation laws is not given on this web page. If you browse through the university of Fla webpages, you'll see more detailed discussion of why (for instance) the transverse electric field gets multiplied by gamma, and the parallel electric field does not.

If you happen by some chance to be familiar with tensors (which is probably unlikely), you'll see that the given equations are just what one expects from a rank 2 tensor - tensors transform in a uniform manner, which is one of the things that makes them tensors.

 

[pervect]

Another possible way to derive the E&M transformation laws is to use the Electromagnetic 4-potential

http://en.wikipedia.org/wiki/Electromagnetic_potential

You'll need to know that E = -\nabla \Phi, and B = \nabla \times \vec{A}

to get the electromagnetic fields out of the 4-potential. (The Wiki article unfortunately doesn't give these important equations, though I think they are given in some of the links).

Knowing that the vector potential (\Phi,c \vec{A}) transforms
as a 4-vector gives you another route for the correct transformation laws for the electromagnetic field.

The "re-calculate from scratch" (harder) approach gives you the Lienart-Wiechart potentials that Hans mentioned. Basically, using the Lorentz gauge, Maxwell's equations reduce to d'Almbert's equation, and the solution for this problem is the Lienard-Wiechart potentials.