# Relative difference in laws of electrodynamics

1. Aug 13, 2012

### nouveau_riche

Consider a train moving at speed 's' and there is a charge particle at rest relative to the observer at train. The second observer on a ground see the charge particle and observer moving relative to him, and infer the existence of a magnetic feild strong enough that its feild is significant at few centimeters from the charged particle. He decides to shoot an arrow with a circular loop hinged on it and there is a LED attach to the loop.
The observer at ground shoot the arrow as he see the train coming near him. The arrow passes near to the charge particle (say a few centimeteres away that he could feel the magnetic influence predicted by observer on ground).

according to the observer on ground the change in magnetic flux from the loop will induce an emf and current will flow, this will light up the LED, whereas from the point of view of observer on train there is no magnetic feild so the LED should not glow

relativity says that the observer in each frame will conclude the same no matter their perception is different to explain the phenomena, but in scenario described above they both come at a different conclusion, where i got wrong(if i did)?

2. Aug 13, 2012

### Staff: Mentor

In the trains frame the changing E field produces a current in the wire through electric induction.

3. Aug 13, 2012

### nouveau_riche

but the observer at ground also knows that the electric effect would add to the magnetic one whereas the observer at train only account for electric induction

also if you presume that at the moment when arrow passes through the effective feild region there is a component of velocity along the velocity of train, the change in electric feild as expected by the observer at train will be less than that as expected by the observer at ground.

Last edited: Aug 13, 2012
4. Aug 13, 2012

### Staff: Mentor

This is correct. Different frames will have different E and B fields, but all experimental measurements will be agreed upon.

5. Aug 13, 2012

### nouveau_riche

what i want to say is that from the reference of observer on ground there must be a magnetic interaction and an electric but from the reference frame of observer on train the interaction should only be of electric kind.

the change of electric field as experienced by the observer at ground is more in magnitude than the change experienced by the observer on train had there been a component of velocity of arrow in the direction of train, therefore their prediction will be different

6. Aug 13, 2012

### Staff: Mentor

Yes, but a particle under an EM force only "knows" the net EM force on it, and it doesn't matter one bit to the particle if different frames disagree about how much of that net force is due to E and how much is due to B.

No, it will not. You are free to work it out quantitatively by yourself if you like, but it can easily be seen simply from the Lorentz covariance of Maxwell's equations and the Lorentz force law, which govern all of classical EM. If you do work it out, do not forget that a moving sensor will be time dilated and length contracted when it makes any measurements.

7. Aug 13, 2012

### nouveau_riche

the speed with which train and arrow moves is negligible in comparison to the speed of light, the effect of length contraction and time dilation would not alter the result much

8. Aug 13, 2012

### Staff: Mentor

You would be surprised. The EM interaction is so strong that even very small length contraction and time dilation effects become measurable. In fact, that is the whole basis of the relativistic explanation of magnetism, which is easily measurable even with drift velocities on the order of tenths of a mm per second.

9. Aug 13, 2012

### nouveau_riche

could you suggest me a link where i can find an experiment that measures those small changes

10. Aug 13, 2012

### Staff: Mentor

An interesting historical note: This particular asymmetry is the one that Einstein chose to motivate his discussion of SR. The first paragraph of his 1905 paper provides the problem statement:

11. Aug 13, 2012

### Staff: Mentor

Any experiment involving the magnetic field of a straight wire is a good example. See the explanation in the link for the theory, particularly the section "Magnetism as a consequence of length contraction".

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

Other good experiments would be to measure the EMF induced from moving a loop past a magnet or moving the magnet past the loop at the same speed. EDIT: Nugatory already mentioned this kind of experiment.

Last edited: Aug 13, 2012
12. Aug 16, 2012

### ApplePion

Consider very carefully the configuration of the magnetic field in the frame where the charge is moving. It does not seem that such a configuration will have a net flux thru the coil. Do it carefully.

13. Aug 16, 2012

### ApplePion

DaleSpam: <<In the trains frame the changing E field produces a current in the wire through electric induction.>>

In the train's frame, the charge is perfectly at rest and is just producing a non-changing Coulomb field.

You seem to want to think there is a B field in the frame where the train is moving, and somehow then convert back to the frame where the train is at rest. But one can evaluate the situation in the rest frame of the train very directly--in that frame the charge is simply at rest, minding its own business, producing an unchanging Coulomb field.

14. Aug 16, 2012

### Staff: Mentor

Sorry, looking back I can see that I worded it poorly.

I am aware that there is no B field, and that the E field is the standard static field given by Coulomb's law. I meant that the E field in the wire is changing over time as the wire moves over time through the spatially varying E field. This causes a current through electric (not magnetic) induction.

15. Aug 17, 2012

### ApplePion

I've changed my mind. In my example I only considered the situation where the normal to the loop points in the direction of the charge, not the general case.

I now think this is a very interesting and non-trivial problem.

16. Aug 17, 2012

### ApplePion

DaleSpam: "I meant that the E field in the wire is changing over time as the wire moves over time through the spatially varying E field. This causes a current through electric (not magnetic) induction."

I don't see why the thing you refer to as "electric induction" would cause a current.

For a situation where there is a non-vanishing time derivative of the electric field thru the coil, we need to look at it in a frame where the charge is moving towards the coil. The "curl of B" Maxwell's Equation refers to what is happening at a fixed point. So what happens is the time-varying electric field induces a tangential *magnetic* field around the stationary loop. But this does not produce any force on the charges in the loop! A time varying magnetic field, on the other hand, would produce a tangential E field, and *that* would have produced a force.

Even if you think that the "line integral of the magnetic field is equal to the time derivative of the electric flux" rule applies to flux caused by the loop moving, the magnetic field thus produced interacting with the velocity of the loop will not produce forces in the appropriate direction to cause current to flow thru the loop.

I probably should clarify what I mean by saying that the"line integral of the magnetic field is equal to the time derivative of the electric flux" rule applies to a stationary loop--lots of people, including myself, have been sloppy with this. Maxwell's Equations expplicitly apply to fixed points. However, everything obeys the Principle of Relativity, so there is complementary process. Indeed, what I am going to tell you comes from Einstein's original paper, and appears to have been much of his motivation. Consider a coil with current moving towards a second coil. Both coils are normal to the x axis, and their relative motion will be along that axis. By the "curl of E equation", the increasing magneticc flux thru the second coil will generate an E field which will cause charge to flow. Now consider this in the frame where the second coil is moving and the first coil is at rest. It might seem that in that frame the increased mnanetic flux causes an induced E field, causing current to flow. But *that* is not what is happening. What is happening is that the second coil is moving thru a magnetic field which happens to have components in the y and z directions, and in this frame it is the V x B force that is causing the current to flow, not changing of flux. I suspect some here will think I am wrong, and I would urge those people to read Einstein's original paper--this discussion featured prominently in his argument for the Principle of Relativity being operative for electromagnetism.

17. Aug 17, 2012

### Staff: Mentor

Consider the case where a conductor abruptly moves from a region of 0 field to a region of uniform field. Before the transition the conductor is uncharged, some time after the transition the conductor has a dipole charge. In order to go from uncharged to dipole there must be a current.

18. Aug 17, 2012

### ApplePion

"Consider the case where a conductor abruptly moves from a region of 0 field to a region of uniform field. Before the transition the conductor is uncharged, some time after the transition the conductor has a dipole charge. In order to go from uncharged to dipole there must be a current."

But the geometery of the situation we are discussing will not lead to any net circulation of charge in the loop.

Consider a situation where everything is in the x y plane and there is a positive charge at the origin, and a rectangular loop whose verices are (x = 10, y = 3) (x= 10, y = -3) (x= 20, y =3) (x= 20, y = -3). The loop is moving in the x direction towards the origin. The point charge is moving in the positive y direction.

First ignore the motion of the point charge. As the loop gets closer to the point charge at the origin, the field from that charge will cause more negative charge to be at (x= 10, y = 3) and it will cause more more negative charge at (x= 10 y = -3) --just follow the field lines from the Cololoumb field. But this is not causing a dipole moment--the shifts would need to be opposite at those two points. Same analysis for the other two points. So there is no net circulation of charge around the loop.

If you want to take into account that the point charge is moving, you still won't get net circulation in the loop. Indeed, the actual criterion for net circulation on the loop is that the E field has a curl that curls around the loop. This will not occur for a uniformly moving charge.

Now contrast it with the situation in the frame where the point charge is moving in the x direction and the loop is not. The point charge produces a magnetic field due to its y direction motion; and the flux of this magnetic field increases as the point charges x direction velocity brings it it closer to the loop. So in this frame there *is* net circulation of charge in the loop.

So the electric induction you refer to does not resolve things.

19. Aug 17, 2012

### Staff: Mentor

It doesn't have to be a net circulation in order to be a current. In both frames the current is transient and any illumination of the LED very brief.

That is because your loop is passing around the charge rather than next to the charge. The scenario in the OP, to my understanding, was a loop passing near a charge, which would induce a dipole charge distribution. That is what I was describing above.

I don't think this is correct. With the geometry you have suggested I think that the magnetic flux will not change as the charge goes through the loop. [EDIT: actually I completely misunderstood your geometry, see below.]

Regardless of what scenario you choose to analyze, it is impossible to correctly use Maxwell's equations to predict some experimental outcome in one frame and to correctly use Maxwell's equations to predict a different outcome for the same experiment in another frame. Any configuration which results in any measurement, like a diode lighting, will give that same measurement in any frame. This follows immediately from the invariance of Maxwell's equations under the Lorentz transform.

Last edited: Aug 17, 2012
20. Aug 17, 2012

### pervect

Staff Emeritus
There does , unfortunately, seem to be some misinformation out there, but a short proof of the invariance of Maxwell equation under the Lorentz transform can be found at http://hepth.hanyang.ac.kr/~kst/lect/relativity/x850.htm [Broken]

Key parts of this can also be found in the wikipedia as well, http://en.wikipedia.org/w/index.php?title=Lorenz_gauge_condition&oldid=505739422, though the wiki article discusses the solution in the Lorentz gauge without specifically demonstrating that it's Lorentz invariant.

Any good E&M text should also have this information.

The fundamental idea behind the proof is to first introduce the electric potential V and the magnetic vector potential A, and then impose the Lorentz gauge condition. It can then be demonstrated that in this gauge, A transforms as a 4-vector, which means that it's Lorentz invariant.

I'm not sure if the OP is familiar with 4-vectors or not. 4-vectors are any sort of vector that transforms via the Lorentz transform. See for instance http://en.wikipedia.org/w/index.php?title=Four-vector&oldid=505607435. Griffiths EM textbook and Taylor and Wheeler's "Space time physics" should both mention 4 vectors for the unfamiliar.

So we are left with A being a 4-vector. A close inspection shows that E and B, by themselves are NOT 4-vectors, though they are closely related. They can be thought of as pieces of a bigger tensor, the so-called Faraday tensor.

See for instance http://en.wikipedia.org/w/index.php?title=Electromagnetic_tensor&oldid=505147584, Where the Faraday tensor defined by [itex]F_{ab} = \partial_a A_b - \partial_b A_a[itex] is a rank 2 Lorentz invariant tensor.

To attack these results, one would either have to claim that:

$$\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2}- frac{\partial^2}{\partial y^2}- \frac{\partial^2}{\partial z^2}$$

was not invariant under the Lorentz transform, or that the charge current vector

(rho, Jx, Jy, Jz), the charge-current density was not a 4-vector. If I had to guess where the fundamental confusion arose, I would guess it arose on this point.

Last edited by a moderator: May 6, 2017