Electricity and Magnetism Problem

[AKG]

1. Is Ampère's law ($\nabla \times \mathbf{B} = \mu _0 \mathbf{J}$) consistent with the general rule that divergence-of-curl is always zero? Show that Ampère's law cannot be valid, in general, outside magnetostatics. Is there any such "defect" in the other three Maxwell equations?

I'm not sure what to do at all here. It seems that that, for the first question, I essentially have to prove that $\nabla \mathbf{J} = 0$, but given that $\mathbf{J}$ could be just about anything, I don't know how to do this. Second, I don't know what it means "outside" of magnetostatics. Does this mean when the current at some point in space changes with time? How would I do this? And of course, since I don't understand that, I don't know how to do the third part of the question either.

2. Suppose there did exist magnetic monopoles (electric fields have monopoles, for example a point charge is a monopole, since the electric fields starts from that point and the field lines continue forever, on the other hand, magnetic fields don't start or end anywhere, e.g. the magnetic field around a straight current-carrying wire is just a bunch of circles, and there's no start or end of the circle). How would you modify Maxwell's equations and the force law "F = q(E + v x B)", to accommodate them? If you think there are several plausible options, list them, and suggest how you might decide experimentally which one is right.

No idea what to do here.

 

[whozum]

1. Consider Maxwell's revision of Ampere's Law. The flaw in Ampere's original Law comes in when considering a loop that encloses one plate of a capacitor.

2. No idea. Perhaps F=q(E+vxB+B).

 

[AKG]

What was Ampère's original law?

 

[whozum]

Ampere's Law says $$\int{B}{dl} = \mu_0 I$$

Maxwell's revision, named the Ampere-Maxwell Law says $$\int{B}{dl} = \mu_0 I +\frac{d}{dt} \int{E}{dA}$$