Confusion about the direction of the vectors: motional EMF
B is roughly along the direction of the core, except in the air gaps.
B is irrelevant to this problem (!!!), or at least mostly so.
A better wording for "when the coil passes the air gap" is "as the coil passes through the air gap".  It is the motion of the coil in the air gap that generates a voltage.
The reason that the motion of the coil in the air gap generates a voltage is because as the coil moves through the air gap it cuts across the flux $\phi_3$, getting ever less flux as it moves.  This is why the B field is made irrelevant -- all you need to know is that the coil is seeing all of $\phi_3$ when it's inside the core, and none when it's outside, and the change must generate some voltage.


You figure out the direction of B along the core using the right-hand rule.  Curl the fingers of your right hand in the direction of current flow in the coil, and your thumb points in the direction of B.