Do moving charges really produce magnetic fields? If moving charges produce magnetic fields and since every object has electrons, when we move the object the electrons in it move as well. So why isn't a magnetic field produced when we move an object?
 A: A better picture would be the one in which a rod is placed in some car, then there would be two frames. The first one is a car frame and the other is a lab frame. In the lab frame, the rod is moving while in the car frame the rod is at rest.
Now I'll state the result without getting into proofs that would make the answer long but see Electricity and magnetism by Purcell.
$$\mathbf{E}'_\parallel=\mathbf{E}_\parallel \ \ \ \ \ \ \  \ \ \mathbf{E}'_\perp=\gamma (\mathbf{E}_\perp+\mathbf{v}\times \mathbf{B}_\perp)$$
$$\mathbf{B}'_\parallel=\mathbf{B}_\parallel \ \ \ \ \ \ \  \ \ \mathbf{B}'_\perp=\gamma (\mathbf{B}_\perp-(\mathbf{v}/c^2)\times \mathbf{E}_\perp)$$
where $\mathbf{v}$ is the velocity of $F'$ frame with respect to $F$.


Why isn't a magnetic field produced when we move any object.

If you are in an unprimed coordinate that is moving with respect to the rod, you will measure the magnetic field. The crazy fact is that the field are not invariant when you go from one coordinate to different coordinate.
