Electric field given a time-dependent magnetic field

[Privatecroat]

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.

 

[Dale]

I constructed a simple magnetic field B = b*tz , ... Helmholtz theorem

How fast does the field need to fall off to satisfy the Helmholtz theorem?

 

[Privatecroat]

Ouch, so I seem to have missed the fact that I need some boundary conditions, which would indeed "center" my solution onto the z-axis. It seems like I just took a look at the differential equations and thought to myself "Oh golly, I will just use that fancy thing I read about in my textbook." Alas, no free lunches indeed.

EDIT: Seems I used the theorem on a field that did not satisfy the conditions. I will need to read further into the theorem and its implications, but thanks for pointing out the error of my ways.

 

[Dale]

No problem!

By the way, it isn't wrong to use that equation as an approximation for some region wherethe field is approximately linear as described. But because you are looking at that small section one of the consequences is that the E field is not unique. You then would need to look further away, to your boundary or symmetry conditions to fully solve for the E field.