Difference between flux of wire loop and wire in a uniform magnetic field

How come when a wire loop sweeps out an area in a uniform magnetic field and moves with constant velocity there is no induced current because there is no change in flux. However, when a straight wire segment sweeps out an area by moving perpendicular to the field(same orientation as loop), there is a change in flux. Why is this? Why does the right hand rule for lenz' law work in this case?

• The loop area doesn't change, but the moving wire segment changes the enclosed flux area by $dA/dt=l*v$. – CuriousOne Jan 11 '16 at 22:49

You have a misunderstanding of flux. Recall that flux is given by $$\Phi=\iint \vec{B}\cdot d\vec{a}$$. Note that there is a necessary differential amount of area that must be considered. Therefore, since a straight wire has no area through which the flux can change, there will be no induced emf.
The reason that there is an emf in the wire is by motional emf. This is deduced from the lorentz force law $$\vec{F}=\vec{E}+q(\vec{v}\times \vec{B})$$. Note that there are free electrons in the wire, each one of which following this law. Lenz's law does not apply here because Faraday's law does not. The right hand rule you use to see where the electrons will go is nothing more than that defined in the vector(cross) product.