Question

How a charge in acceleration produces a magnetic field

Answer 1

If you know basic electrostatics 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!