Which force causes eddy current in a perfect conductor / superconductor plane? As far as I know,

*

*Electric and magnetic field inside a perfect conductor plane are zero (it can be proved by combination of Ohm's law and Faraday's law of induction).

*When there is a magnetic field, eddy currents must exist inside a superconductor to make the net magnetic field zero inside the plane.

*The eddy current flows in nonlinear paths so a force must be applied to the charges to make this movement.

*Lorentz force formula includes all possible forces to an electric charge ($F=qE+qvB$)

*In our case, $E$ and $B$ are zero, so the Lorentz force must be zero.

So which force is applied to the electric charges and causes their movement (eddy current) in a perfect conductor plane?
 A: It is still the electromagnetic (Lorentz) force. A couple of your assumptions are wrong.
First, in a superconductor no force is required for a steady current to exist, even a circular or more complicated "bent" current. You should not think of the electrons forming a superconducting current as little classical point particles racing around and accelerating as they turn. They are not spatially localized very strongly and they come in pairs which "balance" each other out. There is no force involved in a steady superconducting current.
Second, magnetic fields do penetrate a short way (the London depth) into a superconductor. This is called the skin effect in normal conductors, and it is slightly different for a superconductor than a normal conductor but it still exists.
So, in short, your assumptions 1 and 3 are not correct. There is a magnetic field down to the London depth, and no continuous force is needed to maintain a steady current in a superconductor. The magnetic field down to the London depth allows the surface currents to form, and once formed they need no further force to continue.
