Given an electric field in vacuum:
Find the magnetic field under condition that |B| -> 0 as t -> infinity.
I clearly have to use Maxwells equations to obtain the electric field here. Our equation for B has to satisfy them. More specifically Faraday's law and Ampere's law I think are most important here.
The Attempt at a Solution
I am going to use Faraday's law, take the - curl of the electric field and integrate it w.r.t to time. Curl of E will be E_0/(c*t^2) ( 1, 0 , 0).
Integrate this with a minus sing w.r.t to time, and get E_0/(c*t)(1,0,0) + F(r), where F(r) is a function of r we need to find.
Now, we also need B to satisfy Ampere's law and vanish at infinite time. But this field has zero curl, unless F(r) is non-zero, but how can F(r) be nonzero if B has to vanish at infinity? And even if it was non-zero, I have no idea how it could satisfy Ampere's law, as dE/dt if a function of t, but curl of B is not. And I can not find any other form of B except E_0/(c*t)(1,0,0) + F(r) as nothing else seems to satisfy Faraday's law. I am entirely confused.