Is current in a wire frame-specific?

[DJ_Juggernaut]

There is current flowing in a wire. An observer at rest with respect to current, says there is a magnetic field (B) around wire, due to current, I. [Biot-Savart law].

Now switch to someone moving at the speed of current. The current appears stationary to this someone. Therefore this someone says, I = 0, and therefore, magnetic field B = 0. Hence Biot-Savart law does not apply in this case. So goes the mainstream interpretation.

But I question this because for someone moving at the speed of current, the protons move in the opposite direction, hence, current I is not zero. Therefore B is not zero. Thus Biot-Savart law still applies to this someone.

My question is, is current in a wire frame-specific or not?

 

[pervect]

The current can't be simultaneously zero and non-zero. There is a frame in which the current is zero, in this frame positive charges (usually associated with atoms) and negative charges (usually associated with electrons) are moving in opposite directions. If it weren't for relativistic effects, the density of charges would be frame independent and the velocities would be equal. Relativistic effects make the issue a bit more complex.

Assuming that there is no frame in which the current is zero just doesn't make much sense. There seems to be some disconnect here in how you are calculating current, perhaps you are ignoring the contribution of the positive charges in the wire to the current in a frame where the wire isn't at rest?

 

[Orodruin]

There is a frame in which the current is zero

This is not true unless the wire is charged such that the charge density is larger than the current density. If this is not the case the 4-current is space-like and there is no frame with zero current.

An observer at rest with respect to current

There is no such thing as ”being at rest wrt the current”. You might be meaning with respect to the wire.

Now switch to someone moving at the speed of current. The current appears stationary to this someone. Therefore this someone says, I = 0, and therefore, magnetic field B = 0. Hence Biot-Savart law does not apply in this case. So goes the mainstream interpretation.

Again, there is no such thing as wrt the current. There is no frame where the current is zero unless the conductor is charged.

Regardless, if you just consider a line of moving charges, the Biot-Savart law would still be correct - the magnetic field would be zero in that frame. The electric and magnetic field mix under Lorentz transformations.

But I question this because for someone moving at the speed of current, the protons move in the opposite direction, hence, current I is not zero. Therefore B is not zero. Thus Biot-Savart law still applies to this someone.

Yes. This is the standard application of the theory.

 

[DJ_Juggernaut]

The current can't be simultaneously zero and non-zero.

If I am not mistaken this "is" the mainstream interpretation of current in a wire. If you ride along with current, the current is stationary. Therefore, to you, I = 0. Therefore B = 0.

 

[Orodruin]

If I am not mistaken this "is" the mainstream interpretation of current in a wire. If you ride along with current, the current is stationary. Therefore, I = 0. Therefore B = 0.

No. This is definitely not the mainstream ”interpretation”. It only holds for a line of charges that is moving, which has net charge unlike a neutral conductor.

 

[Dale]

If I am not mistaken this "is" the mainstream interpretation of current in a wire. If you ride along with current, the current is stationary. Therefore, to you, I = 0. Therefore B = 0.

The charge density, $\rho$, and the current density $\mathbf j$ together form a four-vector called the four-current $(\rho,\mathbf j)$. The norm of that four-vector is invariant, $-\rho^2+\mathbf j^2$. So if that invariant is positive in one frame then it must be positive in all frames.

Under the usual setup where the current carrying wire is uncharged in the lab frame that invariant is clearly positive, and therefore there is no frame where $\mathbf j = 0$

 

[DJ_Juggernaut]

There seems to be some disconnect here in how you are calculating current, perhaps you are ignoring the contribution of the positive charges in the wire to the current in a frame where the wire isn't at rest?

Current is a flow of charge(s). What is the disconnect here? If you ride along with them, at the speed of current, then the current is stationary to you. That is, I = 0. This I believe is the mainstream interpretation of current. See quote below.

Feynman in a lecture: Because they are moving, they will behave like two currents and will have a magnetic field associated with them (like the currents in the wires of Fig. 1–8). An observer who was riding along with the two charges, however, would see both charges as stationary, and would say that there is no magnetic field.

 

[Dale]

at the speed of current

You have been told already that there is no "speed of the current". There is a speed of the charge carriers, but that is different. The same current can be achieved by a small charge density moving fast or a large charge density moving slow.

 

[Orodruin]

Current is a flow of charge(s).

A current is a flux of charges. This means that current is a measure of how many charges pass per unit time, but the same current can be due to a few slowly moving charges or many fast moving charges.

Edit: I guess the refresh button would have had some use ...

If you ride along with them, at the speed of current, then the current is stationary to you. That is, I = 0.

No. The current due to the charges that carried the current in the rest frame of the wire might be zero, but that does not mean that the current is zero because in order for the wire to be neutral there must be opposite charges that will be moving in your new frame.

Feynman in a lecture: Because they are moving, they will behave like two currents and will have a magnetic field associated with them (like the currents in the wires of Fig. 1–8). An observer who was riding along with the two charges, however, would see both charges as stationary, and would say that there is no magnetic field.

You seem to be misinterpreting Feynman. It seems as if he is talking about two particular charges here, not a current. There are no opposite charges to make the conductor overall neutral.

 

[DJ_Juggernaut]

You seem to be misinterpreting Feynman. It seems as if he is talking about two particular charges here, not a current. There are no opposite charges to make the conductor overall neutral.

Ah, an important detail I missed.