Current and stationary charges

[gumthakka]

I've learned that moving charges produce magnetic fields which in turn affect other charges in motion. After seeing explanations that point to special relativity, I am kind of confused. Can **ALL** magnetic fields be accounted as some kind of electric field from a particular reference frame?

And if there is *relative* motion between the electrons of the wire and the charge at rest(from the lab frame), then will it not experience a magnetic force from the electron's reference frame? I am not sure if that is the actual case, so even if the stationary charge **is attracted** to the wire, can it be accounted as an electrostatic force from the lab frame due to length contraction and as a magnetic force from the electrons POI?

I am not even completely clear with even how to phrase the ambiguity I have in my mind. Detailed answers are very much appreciated :)

 

[A.T.]

Summary:: Why don't current-carrying wires affect a stationary charge placed near them? Will the electrostatic force not come into play?

Not if the wire is neutral overall.

 

[Meir Achuz]

The moving electrons are moving through fixed positive ions, so the net charge at any point is zero.

 

[Dale]

Summary:: Why don't current-carrying wires affect a stationary charge placed near them? Will the electrostatic force not come into play?

Can **ALL** magnetic fields be accounted as some kind of electric field from a particular reference frame?

No. $E^2-B^2$ is an invariant of the electromagnetic field (in units where c=1). So if that invariant is negative then there can be no reference frame where $B=0$. Such a field cannot be described merely as an electric field from a different frame.

However, for a single charge it is always possible to use its rest frame to get rid of the magnetic force. The magnetic field is still present, and necessary for describing the force on other charges.

 

[Vanadium 50]

Why don't current-carrying wires affect a stationary charge placed near them?

They do.

Will the electrostatic force not come into play?

Why would there be an electrostatic force on an uncharged wire?

 

[gumthakka]

They do.
Why would there be an electrostatic force on an uncharged wire?

What I meant was, how would one account for the change in charge density caused by length contraction when charges are moving in a current-carrying wire? Will it not result in a net electrostatic force on a stationary charge?

 

[jbriggs444]

What I meant was, how would one account for the change in charge density caused by length contraction when charges are moving in a current-carrying wire? Will it not result in a net electrostatic force on a stationary charge?

If we say that the stationary charge is stationary, that means that we've chosen a frame of reference. If the wire has zero net charge per unit length in that frame of reference, the wire has zero net charge per unit length that frame of reference. The motion of the charges does not enter in.

Possibly you are thinking about starting with a neutral wire of infinite length and then inducing a current flow in that wire while maintaining the proper separation between each electron and the next (and thereby changing the wire's net charge per unit length as an interesting side-effect). But that is not the scenario at hand.

But perhaps we can look at it another way. Instead of an infinite wire let us consider a circuit. We have a battery, a switch and an insulated wire with some finite resistance. We start with the switch open and no current flowing. We close the switch. The charges start moving. Due to length contraction, the expectation is that they move further apart. But if they move further apart, where can they go?

 

[A.T.]

What I meant was, how would one account for the change in charge density caused by length contraction when charges are moving in a current-carrying wire?

There is no change in charge density in the rest frame of the wire. The fields of the moving charges are contracted, but still repulsive, so the charges still spread out as far as possible. The same number of charges in the same length of the wire results in the same distances between the charges.

See the diagram here:
https://www.physicsforums.com/threa...ation-of-electromagnetism.982883/post-6284262

 

[vanhees71]

If you argue relativistically you have to take into account the Hall effect, i.e., the full relativistic version of Ohm's Law, and then the current conducting wire gets charged (in the restframe of the wire) due to this effect (though it's a very small charge density of order $v^2/c^2$, where $v$ is the velocity of the charge carriers making up the current; one should note that for household currents in usual wires $v \simeq 1 \text{mm}/\text{s}$). For details see

https://www.physicsforums.com/insights/relativistic-treatment-of-the-dc-conducting-straight-wire/