Static free charge in a time varying infinite uniform magnetic field

[NavStar]

Hello,

Let's imagine we have an infinite plane (or large enough compared to the region of interest and measurements) pierced in normal direction by magnetic field B which is uniformly distributed but time varying. For the sake of simplicity we'll presume the magnetic induction is linearly (and perpetually) changing in time.

According to Maxwell equations that would generate a "curly" electric field with constant curl vector:

∇×E = -∂B(t)/∂t=const

The question is how can one visualize the actual vector field of E (the solution for E). Are there infinite number of solutions?
The traditional way of visualizing electric field lines curling around the magnetic field lines seems to fail in this case (which B lines should the E lines wrap around?).

And the consequence question - if we place a static free charge in the field described - how will it start moving (in what direction), what would be it's trajectory having in mind that once accelerated by the E field it will start being affected by the B field (I assume Lorentz low would account completely for both the effects).

 

[Cryo]

I am not sure you have desribed a physically viable situation. Uniform magnetic field will have no divergence and no curl. By Helmholtz decomposition such field is zero
(https://en.wikipedia.org/wiki/Helmholtz_decomposition).

We can assume that magnetic field is nearly constant with slight decay away from the centre. This gives you nonzero curl of magnetic field, and therefore non-zero first derivative of electic field (since $\boldsymbol{\nabla}\times\mathbf{B}=c^2\mathbf{\dot{E}}$), but then electric field must be at least quadratic in time.

This however creates a problem since $\boldsymbol{\nabla}\times\mathbf{E}=-\dot{B}\mathbf{\hat{z}}$, and you said that magnetic field is linear in time (so its derivative with respect to time cannot be quadratic in time).

Of course it is possible to create a region of nearly uniform magnetic field that is changing in time. However, I suggest you get there by starting from a true solution of Maxwells equations. For example a standing wave in the cavity will give you a region of nearly uniform oscillating magnetic field (which will be roughly linear in time for a short period).