We can generalize the concept of motional emf for a conductor with any shape, moving in any magnetic field, uniform or not (assuming that the magnetic field at S each point does not vary with time). For an element d l of the conductor, the contri- bution dE to the emf is the magnitude dl multiplied by the component of v X B (the magnetic force per unit charge) parallel to d l ; that is, d E = (v X B)dlcostheta For any closed conducting loop, the total emf is the integral.

"This expression looks very different from our original statement of Faraday’s law which stated that emf = -dmagnetic flux/dt. In fact, though, the two statements are equivalent’'

What I don’t understand is that according to faradays law if there is no change in magnetic flux in the closed loop then the induced current is 0. However suppose that we have a closed conducting loop moving through a uniform magnetic field. Faraday’s Law tell us that the induced emf is 0 since there is no change in flux. However according to the first statement there should be a motional emf created as the loop moves through a uniform magnetic field. This is contradictory anyone knows what went wrong ?

If the field is uniform and the velocity is constant then we can pull $$\mathbf v\times\mathbf B$$ out of the integral.
$$\mathcal E=\oint_C(\mathbf v\times\mathbf B)\,\cdot\text d\mathbf l=(\mathbf v\times\mathbf B)\cdot\oint_C\text d\mathbf l$$
Since $$\oint_C\text d\mathbf l=0$$, $$\mathcal E=0$$, so there is no contradiction.