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 isn't any change in either the strength of a magnetic field or the area perpendicular as the conductor moves through the magnetic field.
Why is this incorrect?
You can adapt the wording of Faraday's law to take in your case of the straight wire moving in the magnetic field.
$$\mathscr E=-\frac{d\Phi}{dt}.$$
Here you regard $\frac{d\Phi}{dt}$ as the rate of cutting of flux (flux swept through by the wire per unit time). The equation follows from the magnetic Lorentz force, $q\ (\vec v \times \vec B)$ on the wire's charge carriers as they move with the wire, as Señor O points out. One approach is to say,
emf = Work done per unit charge on a charge carrier, $q$, going the length of the rod. So
$$\mathscr E =\tfrac 1q q\ (\vec v \times \vec B).\vec l=(\vec l\times \vec v).\vec B=\text{(directed area swept by rod per second }.\vec B)=-\frac{d\Phi}{dt}.$$
If you want to think about this situation in terms of Faraday's Law (which in integral form applies to any area in space), you get to pick the area to consider. Some choices of area will be useful and others won't. A useful choice is the rectangle whose right edge moves with the conductor, but whose left edge is stationary. This rectangle is bounded by the dashed lines, the moving conductor, and the not-moving segment labeled $l$. As the conductor moves, this rectangle grows in area, enclosing more and more of the magnetic field. Thus the flux through the rectangle increases (into the page) and so Faraday's Law says an emf is generated in the counterclockwise direction around the border of the area.
You should be able to convince yourself that you get the same effect in the conductor if you choose a shrinking rectangle whose left edge is the conductor.
This is just the Lorentz force law.
Charged particles in the conductor will feel a force $\vec{F} = q\vec{v} \times \vec{B}$. This scenario (dragging a conductor through a magnetic field) is referred to as the Hall effect.
Setting emf = change in flux is a nice shortcut for well-defined current loops, but if you're not dealing with a well-defined current loop, don't use it.
