Why do these calculations of EM fields for a magnet and wire loop seem inconsistent?

Question

Suppose you have a conducting circular wire loop and a magnet moving towards each other. They move along the $z$ direction with nonrelativistic constant speed $v$. Let the $B$ field of the magnet in its reference frame be parallel to the $z$ direction: $$\vec{B} = (0,0,-z),$$ so the strength of $\vec{B}$ linearly decreases along $z$. Let the surface normal of the area enclosed by the wire loop be pointing in the $z$ direction too. Now here is my problem: If I'd like to calculate the fields and forces acting upon charges in the wire, I get different results.

In the magnet's reference frame, there is a static magnetic field $\vec{B} = (0,0,-z)$, and the wire loop is moving with a constant velocity $\vec{v} = (0,0,1)$ towards the magnet. In this case there is no $E$ field, so the resulting force on charges is $\vec{F} = q\vec{v}\times\vec{B}$. Since $\vec{v}$ and $\vec{B}$ are parallel, the cross product is 0, so there is no force.

If I look at the wire's reference frame, there is a time-varying magnetic field producing an electric field. Here's where I don't quite get it. The electric field will be $$\vec{E}' = \vec{v}\times\vec{B}$$ according to Wikipedia (gamma is aproximately 1). In this case $$\begin{align} \vec{E}' &= 0 \\ \vec{v} &= 0 \\ \vec{F} &= q(\vec{E} + \vec{v} \times \vec{B}) = 0 \end{align}$$ just like in the other reference frame.

But if I try to use Faraday's equation instead, the wire loop sees a time-varying magnetic field, with a non-zero surface integral, so there has to be a non-conservative, non-zero $E$ field acting upon the charges.

I've tried this with different magnetic fields and every time my calculations fail miserably. I'm missing something somewhere, but I don't know why.

Answer 1

You're missing the fact that your situation as you've defined it is physically impossible. Remember that one of Maxwell's equations is $\vec{\nabla}\cdot\vec{B} = 0$, but your magnetic field configuration $\vec{B} \propto (0, 0, -z)$ doesn't satisfy this. Given that, it's not surprising that Maxwell's other equations give inconsistent results.

Incidentally, $(0, 0, -z)$ is not the magnetic field of a point magnet. You would need some extended, continuous distribution of magnetic charge to get that field. If you did assume that magnetic monopoles exist and you can have them distributed in a way that produces this field configuration, you'd have a "fluid" of magnetic monopoles in motion in the wire's reference frame, which would account for the electric field.