Electromagnetism problem: where does the magnetic field come from?
Consider the following problem:
Consider a plane with uniform charge density $\sigma$. Above the said plane, there is a system of conducting wires made up of an U-shaped circuit on which a linear conductor of lenght $d$ can slide with constant velocity $v$. The system as a whole has a rectangular shape and is parallel to the plane. (See the picture). Calculate the line integral of the magnetic field $\bf B$ along the perimeter $L(t)$ of said rectangle as a function of time.
My professor solves this problem using <PERSON>'s fourth equation in integral form, assuming that the current density $\bf {J} $ is everywhere null, and that the electric field $\bf E$ is the one generated by a uniformly charged plane, i.e. 903
perpendicular to the the plane and of norm $E=\frac{\sigma}{2\epsilon_0}$; thus yielding $$\oint_{L(t)} {\bf B}\cdot dl=\mu_0\epsilon_0\frac{d}{dt}\int_{S(t)} {\bf E}\cdot dS=\mu_0\epsilon_0Edv=0.5\mu_0\sigma dv$$
I think there are some things wrong both with this solution:
1. There should be no magnetic field at all! A uniformly charged plane only produces an electrostatic field. (I know there could be a magnetic field generated by the current inside the wires, but then you couldn't assume that $\bf J$ is null everywhere as my professor did!)
2. <PERSON>'s fourth equation does not hold in that form if the domains of integration are allowed to vary with time. In fact, by resorting to the differential forms, we find that plugging ${\bf J}=\vec 0$ and $\frac{\partial {\bf E}}{\partial t}=0$, as my professor assumed, yields $rot{\bf B}=0$, and thus the line integral of the magnetic field over any closed curve, at any istant, should be zero by <PERSON>' theorem!
Therefore, my question is the following.
Are the my professor's assumption ($\bf J$ $=\vec 0$, $\frac{\partial{\bf E}}{\partial t}=\vec 0$) correct, or do both $\bf J$ and $\bf E$ need be modified so as to account for the charges present in the circuit? Is there a current in the circuit at all? 903