How is there a change in magnetic flux from a moving conductor through a uniform magnetic field? My textbook states that when a straight conductor with a velocity perpendicular to itself and the magnetic field (see image), an emf will be induced between the two ends of a conductor. This means that there is a change in magnetic flux, but how is that so? According to the definition of magnetic flux = B*A, there is no change in either the strength of a magnetic field or the area perpendicular as the conductor moves through the magnetic field.
Could someone please let me understand why this is incorrect. 

 A: 
straight conductor moves perpendicular to itself 

I don't quite get what you mean by this. But I think I understood the scope of your question.
In an ideal conductor it has an "infinite" number of free charges that can move. When a charge moves in the presence of a magnetic field it experiences a force given by $\vec{F_{mag}}^{\,}=q\times\vec{B} \, \,$, so this causes the the free charges to move (and this example it happens that the orientations will have the free positive charges move upward to one end). When the free charges do move, you have a difference in potential at the ends of the rod which qualifies as EMF.
A: There is no change in magnetic flux, as you've said. What you're missing is, there is also no EMF induced in any loop. Try drawing a loop you think there is an EMF around. If a positive charge starts at the top of the conductor, goes to the bottom of the conductor, and then goes back to the top, it doesn't gain or lose kinetic energy. I think your confusion can be cleared up by remembering what the Biot-Savart law actually says: It says that changing magnetic flux through a loop results in an EMF around the loop. You've said there's no changing magnetic flux through any loop you can draw, and if you think about it there is also no EMF around any loop you can draw.
