Quote by Meir Achuz
A time varying magnetic field will have time varying vector potential
[tex]\frac{\partial{\bf A}}{\partial t}[/tex] that can exist beyond the field, and induce an E field.

Only in areas where there is a changing magnetic field.
Quote by elfmotat
However, (assuming ∂B/∂t isn't zero) the electric field induced will also be nonzero outside of the solenoid.

∂B/∂t ≠ 0 implies that there is a magnetic field (apart from some specific points in time maybe).