Consider a rectangular slab of permanently magnetized material. The slab's dimensions are $L_x$, $L_y$, and $L_z$, and the slab is uniformly magnetized in the $\hat{x}$-direction. The slab is not accelerating or spinning. Does the slab generate an electric field?

In a frame where the magnet is stationary, we know $\mathbf{E}$ is zero everywhere. In a frame where the magnet is moving, there are at least two ways to attack the problem:

1. Drop $d\mathbf{M}/d t$ into Maxwell's equations, solve for $\mathbf{E}$ and $\mathbf{B}$
2. Solve for $\mathbf{B}$ in the slab's rest frame, and use a relativistic boost to transform $\mathbf{B}$ in the slab's frame to $\mathbf{E}$ and $\mathbf{B}$ in the frame where the magnet is moving.

Both these methods give the result that $\mathbf{E}$ is nonzero in a frame where the magnet is moving.

Now let's consider a long, thin slab ($L_x, L_y \ll L_z$). In a frame where the slab is moving in the $\hat{z}$-direction, is there an electric field (external to the slab) near the 'center' of the magnet? Both the $d\mathbf{M}/d t$ argument and the Lorentz-boost argument seem unchanged. The magnetic field external to the slab does not vanish near the center of the slab, suggesting there is a nonzero electric field.

With the backstory laid out, here's my real question: In a frame where the slab is moving in the z-direction, is there still an electric field in the case where $L_z \rightarrow \infty$?

The Lorentz-boost argument seems unchanged, and suggests that there is. However, in the $L_z \rightarrow \infty$ case, $d\mathbf{M}/d t = 0$, suggesting no electric field. Can this case be calculated without Lorentz boosts? How do Maxwell's equations account for moving permanent magnets in the case where $d\mathbf{M}/d t = 0$?