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We all understand that moving charges create uniform magnetic fields. Saying this, if we consider the example of a neutral current carrying wire, then there is a uniform magnetic field around it. My questions are:

  1. The conductor as a whole is neutral so why would it produce a magnetic field?(I mean there aren't any net charges moving around)
  2. If we take a neutral wire and begin to move it (physically, no current in the wire), will it still produce a magnetic field?(I know it sounds silly but I want to know why this idea doesn't work?)

Any help is appreciated!

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One way to think about this is to consider one current $I_E(t,x)$ produced just by the electrons and another current $I_P(t,x)$ produced just by the protons. Both currents may depend on time and location. In the case of a non-moving neutral wire carrying an ordinary electric current, we have $I_E(t,x)\neq 0$ at least somewhere (because the electrons are moving) and $I_P(t,x)=0$ everywhere (because the protons aren't moving), so the net current is $$ I_E(t,x)+I_P(t,x)\neq 0 $$ at least somewhere, so it produces a magnetic field. The net current is what matters for the magnetic field.

In the case of a moving neutral wire that is otherwise not carrying any current, both $I_E(t,x)$ and $I_P(t,x)$ are non-zero, because the electrons and protons in the wire are both moving with the mechanical motion of the wire. But the electrons and protons are moving together, and since they have opposite charges, this implies $I_E(t,x)=-I_P(t,x)$. Therefore, the net current in this case is $$ I_E(t,x)+I_P(t,x)=0. $$ The net current is what matters, so there is no magnetic field in this case.

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  1. Although the conductor is neutral, charges are of course moving around. The current due to the charges will create a magnetic field. However, the electric field will be zero, as the conductor is neutral (and has no local charge densities).

$\textbf{Note}:$ In the frame where the current carrying charges (electrons) are stationary, the charge density of positive (proton) and negative charges (electron) will not be same, and the conductor will be charged (this is a relativistic effect). The electric field of this charged conductor, when appropriately transformed to the initial frame, will behave as the magnetic field produced by the current in the conductor.

  1. Since the conductor is neutral, and is not carrying current (the current due to protons end electrons will cancel each other), there will be no electric and magnetic field.
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  1. The conductor as a whole is neutral so why would it produce a magnetic field?(I mean there aren't any net charges moving around)

In a neutral current-carrying wire, the charges are balanced. This means that the wire does not produce an electric field. However, a magnetic field is still induced due to the presence of flow of charges.

A magnetic field is produced only when a charged particle is moved. Let us first consider a charged particle, say, an electron in its rest frame. It would only be noticed to produce and electric field, and no magnetic field at all. However, should we change our reference frame to something that is moving with relative velocity to this electron, say another electron, it would not only be observed to be accelerating, but undergoing rotation (a form of acceleration)!

Now to account for the rotation, it is posited that in the second frame of reference from another electron, there is a magnetic field, in addition to the electric field, that rotates the charge.

Having no net charges moving around simply means that there is no electric field.

  1. If we take a neutral wire and begin to move it (physically, no current in the wire), will it still produce a magnetic field?(I know it sounds silly but I want to know why this idea doesn't work?)

There is a difference between a neutral wire and a wire with no current. Look back at the definition of current, defined as the flow of electric charge. In this case, there is no potential difference, hence no flow of electric charge. Hence, moving a wire with no current is impossible to produce an electric field nor a magnetic field.

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