I How relative is the magnetic field of a current carrying conductor?

[lonely penguin]

According to the special theory of relativity the electric and magnetic fields observed from a charge distribution should depend on the relative motion of the observer. For example consider a linear arrangement of point charges which are fixed relative to each other. An observer who is stationary relative to the charges should only observe an electric field (as derived for example using Gauss's law). However if the observer is moving with a relative velocity with respect to the charge distribution (for example in a direction parallel to the stationary charges) then in addition to the electric field a magnetic field will also appear (Gauss’s and Ampere’s law).

My question considers the opposite situation. The magnetic field from a current-carrying conductor in the stationary observer's frame is described by Ampere's law. However, for the observer moving with the same velocity as the drift velocity of the charges and in a direction parallel to the conductor, the moving electrons would appear stationary, leading to the following implication: if the electrons seem stationary, and assuming the positive charges (ions in the conductor) also appear stationary to this observer, no current is observed, and thus no magnetic field due to current is perceived by this observer.

Is this actually possible? Moving a magnetometer with a few mm/s along a current-carrying conductor will create changes in the measured magnetic field?

 

[PeterDonis]

if the electrons seem stationary, and assuming the positive charges (ions in the conductor) also appear stationary to this observer

This can't be the case if there is current in the conductor. You recognize this in the same sentence, but you don't appear to realize that this contradicts your original specification that there is current in the conductor. You can't have it both ways.

If it is actually true that there is a frame in which both positive and negative charges are stationary, then there cannot be a current in any frame. In a frame in which the charges are moving, both sets of charges are moving with the same velocity, and hence there is still no current because the two charge flows cancel each other out.

In fact, in that scenario (i.e., if there is a frame in which both sets of charges are stationary) there is no EM field at all, since the charge densities also cancel in any frame, so the EM field source (charge-current four-vector) is zero. Your original specification was for a scenario in which there is an EM field and hence a nonzero EM field source.

I suggest that you rethink the problem in the light of the above.

 

[Ibix]

assuming the positive charges (ions in the conductor) also appear stationary to this observer,

If there's a current then the electrons are moving relative to the ions, and at least one species is moving in any frame. If there exists a frame where both electrons and ions are stationary then there is no current, and this will be true in all frames.

Moving a magnetometer with a few mm/s along a current-carrying conductor will create changes in the measured magnetic field?

Generally, yes, two electromagnetic field sensors in relative motion will read different things. If you know how to transform a tensor, write down the Faraday tensor (also called the electromagnetic tensor) and transform it, then read off the components in the new frame. Remember to transform the coordinates in the expression for the field.

 

[lonely penguin]

"if the electrons seem stationary, and assuming the positive charges (ions in the conductor) also appear stationary to this observer"
You are absolutely right .... the above sentence is wrong. I actually misprinted this and I only wanted to consider an observer that is traveling along the conductor with the same drift velocity as the electrons. But you have answered my question. If the electrons appear stationary then the ions will not. So there will always be a current. I guess there will be a relativistic effect but it will be negligible for small velocities.
Thanks for the reply

 

[Ibix]

I guess there will be a relativistic effect but it will be negligible for small velocities.

Actually, all electromagnetic phenomena are relativistic. In fact, analysing this case won't make sense unless you take into account length contraction of the distance between the moving species.