Direction of Induced Electric Field

If the magnetic field has boundaries, then the induced electric field will form loops around the magnetic field. The size and shape of the loops will depend on the size and shape of the magnetic field and they will be centered on the effective center of the magnetic field.

Faradays Law in differential form states:

$$nabla times E = frac{- partial}{partial t}B$$

Since the definition of a conservative vector field $V$ is $V=nablaphi$ for some $phi$, and we know that $nabla times (nabla phi) = 0$ for all $phi$, we can state that a conservative vector field $V$ must obey the condition $nabla times V = 0$.

So therefore all electric fields are conservative, unless there is a time-varying magnetic field present.

From here we want to find $E$ given that we know its curl, and there is not a unique solution for this unless we have initial and boundary conditions for our magnetic and electric field. A simple way to think about this; if I just told you that $frac{- partial}{partial t}B =0$ can you use that to tell me $E$ uniquely? No. We need to know its I.C. and B.C. which would be dictated by charges, conductors, etc...