# Why do Moving Charges Produce Magnetic Fields?

*Most recent answer: 01/05/2013*

- saurav (age 16)

Ranchi, India

Hi Saurav,

If you know basic electro*statics* and basic special relativity, then you can understand why magnetic forces are necessary in order for the laws of physics to be consistent.

For example, consider sitting in your lab frame with a neutral wire, consisting of an infinite line of positive charges moving to the right, and another of equal-but-negative charges moving to the left. This configuration has a net current to the left, but since the total charge at each point in space is zero, there is no electric field. So, if an electron is sitting next to the wire, it will not feel a force from the wire, and it will not accelerate.

Now, special relativity says that the particle will also not accelerate if viewed from a rocket moving past the line to the right at constant speed V. Special relativity *also *says that, in this frame, the relative motion of the charges and the rocket causes the negative charges moving to the left to be **length-contracted**, while the positive charges moving to the right will be **length-expanded**. This result comes simply from the relativistic law of velocity addition and the law of length contraction.

So, in any given piece of space in the rocket frame, there is now more negative charge, and less positive charge, than it had in the lab frame. This charge excess will repel the electron. It seems as though the particle should now feel a net force, which we created simply by switching frames!

This cannot be, however, since relativity says both frames are valid points of view, and should agree on observed events (that is, the electron doesn't accelerate towards or away from the line!). To resolve the paradox, there must be a new force, which only occurs when the electron is moving past a current (as it is, in the rocket frame, but is **not**, in the lab frame). So, we say that currents set up magnetic fields, with which moving charges interact via the Lorentz force.

This argument can be made mathematical; you can find an exact treatment in many undergraduate textbooks on electricity and magnetism. To summarize, the laws of relativity and basic electrostatic forces show that we need a new, motion-dependent force in order for our laws to be consistent.

An interesting thing to note is that a similar argument can be made for other forces, like gravity. In a frame which views an object as moving, the object's mass density increases; therefore, there must be a compensating, "gravitomagnetic" force, which arises from moving massive objects. This effect has also been observed!

Hope that helps!

David Schmid

*(published on 01/05/2013)*

## Follow-Up #1: Why does a solenoid behave like a bar magnet?

- Kunal (age 15)

Kolkata,India

I'd like to reverse the question: why does a bar magnet behave like a solenoid?

David's previous answer gave an idea of why solenoids behave the way they do. Relativity+electrostatics implies that currents flowing the same direction attract. So if you have two solenoids with current going around the same way, if they're end-to-end they attract. If they're side-by-side, they repel because the closest current flows are opposite.

So why do bar magnets act like solenoids? We can't give a full explanation here, just some hints. Each electron itself acts like a little magnet, like a little current loop, for reasons hidden deep within quantum mechanics. We describe elsewhere () now in certain materials these electron magnets tend to line up, so that the net effect is like a big current loop. Then they act like solenoids.

Mike W.

*(published on 06/29/2013)*

## Follow-Up #2: Why are electricity and magnetism unified?

- Mark (age 26)

Dubai, UAE

I've marked this as a follow-up to a related question. The quick answer is:

1. Even in Maxwell's equations, there are terms connecting magnetic and electric fields, so that propagating field waves always contained both, not just one or the other.

2. In special relativity, going from one reference frame to another changes the fields from one type to another. In other words, if you have an electric field but no magnetic field in some frame, if you look in a relatively moving frame there will be a different electric field and now some magnetic field. Likewise if you have a magnetic field but no electrc field in some frame, if you look in a relatively moving frame there will be a different magnetic field and now some electric field.

Mike W.

*(published on 07/17/2013)*