Doubt about the electric field produced by a time varying magnetic field along an infinite cylinder

Question

My class had a doubt about the direction and magnitude of the electric field in the presence of a time varying but uniform magnetic field.

Specifically about the symmetry arguments to determine the fact that the electric field must be tangential about the "centre of the circle", but our doubt is why a particular circle should be considered as opposed to any other circle since the magnetic field is uniform and doesn't depend of the distance to any circle.

Further, if the magnetic field pervaded the entire field, in what direction would the induced electric field be.

We came to the conclusion that the situation must be unphysical, but since both the divergence and the curl of the magnetic field are zero in the latter case we are confused as to which law the situation violates

Answer 1

The short answer is boundary conditions. Maxwell's bulk equations alone are insufficient to determine the fields, you need to specify how the fields behave at infinity. This is because the electric field diverges at infinity. The short answer is that the boundary conditions, while not visible from the constant field alone, will prescribe which symmetries are legitimate. In your case, they will select the correct center of the circle.

A physical setup where your argument would give the correct result would be an infinitely large cylindrical solenoid. In this case, the privileged center is the axis of the solenoid. To get the full solution, you will also need to consider the magnetic field due to the displacement current due to the induced $E$, and continue ad infinitum. For sinusoidal $\alpha$, this will be a Bessel function of the first kind. On a side note, your symmetry argument for (2) is incorrect, translation invariance alone is not enough. You can directly know that $E$ is orthoradial using instead the reflection symmetries about any plane passing by the central axis.

A contrario, a setup where it would not work is to consider two infinitely spaced out plane sheets of opposite currents. In this case, $E$ would be in the same direction as the currents (again by reflection about planes normal to the currents).