I suppose that my main question is how to find the induced electric field given a time-dependent magnetic field, but i will demonstrate my question with an example: I constructed a simple magnetic field B = b*tz , permeating the whole space. The induced electric field will lie in the xy plane (assuming there are no static charges present). Now, from what I understood Helmholtz theorem says that, given divergence (0 in this case) and rotation of a vector field (-b for the electric field in this case) that field is uniquely determined. Then I constructed such a field (E = ½b*yx - ½b*xy). However, then I noticed that any field of shape E = ½b(y+A)x - ½b(x+B)y will also satisfy the conditions (I.E. the centre of the circular field lines will be shifted). Furthemore, why would any point on the xy plane be privileged as the centre of the field lines circulating around it? Now , I could say that the whole space is infact in an infinitely big solenoid, whose axis coincides with the z-axis, and then the electric field lines should circulate around the origin of the xy plane (and all planes parallel to it). In that case, why does my math not check out? Maybe I am missing something obvious, but in any case any help would be appreciated. EDIT: bolded x,y,z are direction vectors only.