Why would a moving infinite region of magnetic field exert an electric force? I'm confused about the Lorentz force in relation to the frame of reference. There are many questions about that here, but I still don't get it. I've tried to break it down to the most basic example.
Lets say we have a single charge moving into the screen with velocity $v$ between two infinitely long magnet bars. The Lorentz force law tells us that the charge will experience a force to the side.

So far so good. But now I change the frame of reference into one where the magnets move at speed $v$ out of the screen and the charged particle remains stationary. The relative motion is the same, so I expect the same force. 
Since the charged particle isn't moving, there is no Lorentz force.
Answers to the other questions explain it with the electric field. Where does the electric field come from in the second frame of reference?


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*The magnets have no net charge, so Gauss' Law can't be it, right?

*The two magnets are now moving along the axis on which they have infinite length, so the motion is not really changing anything observable. They might as well be static. In particular, the magnetic field does not change, so Faraday's law of induction shouldn't create an electric field either.
 A: You can't get away from special relativity, which is what "unifies" electric and magnetic phenomena. The electromagnetic field really is a single field (actually a tensor, but don't worry about that), and its components mix together in the Lorentz transform in a similar way to how the x,y,z components of a usual vector field mix together in a rotation.
There are two ways to find the electric field here: You can perform a kinematic Lorentz transform to obtain the source charges/currents in the moving frame. You can also obtain the electric field component directly from the Lorentz transform of the electromagnetic field.
Lorentz Transform on Sources
An object in motion appears length contracted to observers: $L' = L_0/\gamma$, where $L_0$ is the length of the object its rest frame and $\gamma$ is the Lorentz factor associated with the velocity of the object:
$$ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $$
Consequently, if the object is charged, the charge density increases when in motion: 
$$\rho' = \gamma \rho_0$$
The magnetic field in the rest frame arises from the magnetization $\vec M$ (magnetic dipole density) of the magnets, which is pointed up along the N-S axis. Each magnetic dipole is effectively a small current loop with current $\vec J = \rho \vec u$, where $\vec u$ is the tangential velocity around the N-S axis. 

Then in the moving frame, the speed $u$ is not uniform around the loop, and charge densities at the points of the loop flowing with and against the direction of motion are different:
$$u_\pm = \frac{u \pm v}{1 \pm uv/c^2} $$
$$\rho_\pm = \gamma_\pm \rho_0  = \frac{\rho_0}{\sqrt{1 - u_\pm^2/c^2}}$$
Therefore $u_+ > u_-$, which means $\rho_+ > \rho_-$, so each current loop (magnetic dipole) acquires an electric dipole moment. Each magnetic dipole pointed up, so each electric dipole is pointed to the right. This is the source of the electric field in the moving frame.
Lorentz Transform of E and B fields
If you begin with fields $\vec E$, $\vec B$, and want the fields in an inertial frame moving at relative speed $\vec v$, the new fields are:
\begin{align}
\vec E_{\parallel}' &= \vec E_{\parallel} \\
\vec B_{\parallel}' &= \vec B_{\parallel} \\
\vec E_\perp' &= \gamma(\vec E_\perp + \vec v \times \vec B) \\
\vec B_\perp' &= \gamma(\vec B_\perp - \frac{1}{c^2} \vec v \times \vec E)
\end{align}
In our case $\vec E = 0$ and $\vec B_\parallel = 0$, so the above simplifies to 
\begin{align}
 \vec E' &= \gamma \vec v \times \vec B \\
 \vec B' &= \gamma \vec B 
\end{align}
If the $\vec B$ field is up (from S to N) and $\vec v$ is into the page, this gives an $\vec E$ field pointed to the right.
A: If you think of a magnet as being made up of individual dipoles all aligned with one another (this is what happens in Iron or Neodymium bar magnets), then you can ask "how does each dipole transform under Lorentz transformation?" Under Lorentz transformation magnetic dipoles become electric dipoles. The electric field from the electric dipoles causes the Lorentz force in this frame.
