Will electric field cause electrons to move in disconnected wire? Imagine a straight piece of wire, not connected to anything. Parallel to one of it's ends is a section of circuit with direct current in it. They're placed in such a way, that only about 1/4 (or less) of disconnected wire is near a powered circuit. Question: will electric field in circuit cause at least some of electrons in disconnected wire to move to other end (upper end on picture)? Will this happen, if circuit is a toroidal solenoid, and end of disconnected wire in inserted in hole in the middle of said toroid?
 A: If you had an electric field, charge separation would build up in the disconnected conductor such that the electric field inside the conductor was zero. This movement of charges would constitute a current that very rapidly decayed to zero.
However, neither of the diagrams you show appear have a changing magnetic field because the current in the blue wire appears to be depicted as constant. Therefore no electric field is induced and no current flows in either orange wire.
A: On your left diagram, assuming the straight wire is close enough to the circuit, the magnetic field on the circuit will induce a current in the straight wire due to Faraday induction but only for a brief moment during a d.c. transient (i.e. switch on or switch off of the circuit). For the d.c. steady state of operation, no current will be induced in straight wire since there is no changing magnetic field on the circuit. For an a.c. circuit your straight wire will behave like a reception radio antenna provided the physical length of your wire is similar to the a.c. signal wavelength $λ$ in your circuit with optimum length values for the straight wire for best reception, $λ/2$ or $λ/4$.
For your right diagram, it is known that a toroidal solenoid has no magnetic field outside. Neither around it nor at its cap. All the magnetic field B in contained inside the toroidal solenoid. Therefore, neither a d.c. transient signal nor an a.c. signal will be induced in the straight wire.
As for the electric field of the circuit, for both of the above described cases it is known that for metal conductors electric field is contained inside the wire (or more close to the surface of the wire for high frequency a.c. signals, skin effect) and no electric field is practically radiated outside the wire. Even in d.c. steady state, the wires remain electrostatically neutral due the homogeneous current charge transfer inside a wire.
CAUTION: Electrostatically neutral, that does not of course mean in anyway that touching a naked high powered wire and you being sufficiently grounded, that you will not be electrocuted!
