# Why don't (or why do) current carrying wires attract a stationary charge placed at a distance?

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 :)

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

No. Relativity really tells us that electric and magnetic fields are on an equal footing. In some situations, you can find a frame where there's only an electric field. In others, you can find a frame where there's only a magnetic field. But most of the time, you can't do either.

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'm not sure if this is getting at your confusion, but recall some basic examples in relativity. For example, suppose that in your frame a spaceship passes by you. In your frame, this can happen really quickly because the spaceship is length contracted. In the spaceship's frame, it happens really quickly according to you because your time is dilated. So what's really going on? Is it really time dilation or is it really length contraction? Of course, the point is that the two frames are on an equal footing. Time dilation in one frame can be equivalently described as length contraction in another, and neither is inherently more correct.

Similarly, in some situations, what can be described as a magnetic force due to motion in a magnetic field in one frame, could be described as an electric force due to an electric field in another frame. In each individual frame, absolutely everything works as usual: Maxwell's equations are true, the Lorentz force expression holds, and so on. So, for example, in a frame where a charge is still, it experiences no magnetic force, even if it might in a different frame where it is moving. The description of what is going on changes between different frames, but neither frame is more "correct".

Saying that magnetic forces are "always really just because of electric forces due to a charge imbalance due to length contraction in a different frame" doesn't make sense. It doesn't work in general, and it's kind of like saying "time dilation doesn't really exist, only length contraction does". It's actually the exact opposite of the spirit of relativity.

• My confusion came from considering electric effects to be more "fundamental" than magnetic effects because they are kind of more intuitive to me (because I think of Electric effects originating from the existence of monopoles and charge coming from them, thus more "fundamental") and hence the question, trying to validate magnetic effects as coming from a more "fundamental" effect seen from a different POI. So correct me if I am wrong, in reality, both DO exist and we use relativity as a means of interpretation of the same observed effect? Jun 3, 2020 at 7:07
• @HarishRaju In relativity, neither is more fundamental than the other. In fact, you can run the derivation backwards: you can start with magnetic forces, which fundamentally come from currents by the Biot-Savart law, and prove that relativity means that in a different frame, a magnetic field can become this strange, weird, exotic thing that is called an electric field. So in that picture, the magnetic field would be more "fundamental". But what this is really saying is that both are equally fundamental. Jun 3, 2020 at 7:13
• @HarishRaju I wouldn't say that both "DO exist", because that's a philosophical question. I mean, does the hole in a donut "exist"? Do emotions "exist"? Does society "exist"? What we know for sure is that if you want to describe what electromagnetic fields in relativity, the best description (for the purposes of doing calculations, and intuitively understanding their results) involves treating them as equally as possible. Jun 3, 2020 at 7:15

Forget special relativity.

In the lab frame the wire is charge neutral so has no electric field.

However, in the lab frame there is a current. This current produces a magnetic field.

That means there will be a magnetic field at the location of your test charge. However, you specific a stationary test charge. Recall that the magnetic Lorentz force on a charge is proportional to velocity. So even though there is a magnetic field at your test charge, it feels no force.

That is to say, a charge is NOT attracted to or repelled from a current carrying wire.

edit:

Now let's change the reference frame into frame moving parallel to the current carrying wire (frame might be moving with or against the majority charge carriers). Suppose the current is generated by positive charges moving in the $$+z$$ direction within a lattice of negative charges. Speaking qualitatively a few effects will happen.

(1) In the new reference frame the positive and negative charges in the wire experience some amount of the length contraction. The length contraction will differ between the two types of charges because the + and - charges have a relative velocity. This difference in length contraction leads to a resulting difference in the charge density for + and - charges. This difference in charge density means the wire appears to have a net-local charge in this moving reference frame. I believe This net charge could be either positive or negative depending on which way we've boosted resulting in E-field lines pointing towards or away from the wire. This E-field puts a force on the test charge.

(2) In most boosted frames there will still be a net-current through the wire. This current generates a mangetic field at the location of the test charge. However, in this moving frame, the test charge now has a velocity. The test charge is now moving with some velocity through the magnetic field. If you work out the right hand rule you will see that the Lorentz force on this test charge is either towards or away from the wire.

My claim is that the electric force from the length contracted charge densities and the magnetic force exactly cancel out so that there is no force towards or away from the wire no matter how fast you boost along the current carrying direction. I don't prove it here, but someone with more time could more carefully go through all the math, boosts, negative signs, etc. and find the expected result.

• But lets try to change the frame. Assume electrons moving at $v_1$ velocity towards north direction in lab frame. And you are moving in south direction at $v_1/2$ velocity. So the test stationary charge is now in motion also the electrons in wire are moving at $3v_1/2$ velocity. There should be a force on charge now. (I also think length contraction doesn't work 🙌) Feb 4, 2022 at 23:07
• @PredakingAskboss Why doesn't length contraction work? Yes there will no be a magnetic force on the charge due to Lorentz force. But because the charge densities for positive vs negative charges in the wire are differnet (due to differential length contraction due to their differential velocities) there is also an electric force on the test chage which cancels the magnetic force. Feb 5, 2022 at 1:37
• @PredakingAskboss 1: Relativity says charged moving rods contract and that has an effect on force between the rods. 2: Relativity says charged moving point particles move and that has an effect on force between those particles. Feb 5, 2022 at 5:02