what is motional emf derive an expression for it when a rod is moving perpendicular inside a field

Motional emf

A conducting bar moving through a magnetic field will also experience an induced emf.

This is because the bar has charges that can move in response to a magnetic field. Moving the bar at speed v moves these charges at speed v through the field. These charges feel a force:

F = qv ´ B

If the length L of the bar is perpendicular to the field and the rod is moved in a direction perpendicular to both L and B, the force is F=qvB and deflects the charges to one end of the bar. This builds up an electric field which exerts a force F=qE opposite in direction to the magnetic force. The charges continue to be deflected until the forces balance, when:

qE = qvB, or when E = vB

Assuming the electric field is uniform over the length of the bar, the potential difference between the ends of the bar is:

DV = -EL = -vBL

We usually call this a motional emf, so e = -vBL

This is true as long as v, B, and L are mutually perpendicular.

If the bar is connected to a circuit the bar acts like a battery, which is why a current flows in that situation.

There is actually a measureable potential difference between the wingtips of airplanes coming from this. Although planes move through the Earth's field, which is fairly weak, their high velocity and substantial length induce a reasonable emf.