We say that induced potential in a magnetic field is directly proportional to the rate of change of flux but when a wire moving perpendicularly in a uniform magnetic field at constant velocity it is said that a potential is induced.But there isn't any rate of change of flux . So how does it induce a potential?
You've brought up a great question that touches on the fundamental concept of electromagnetic induction. The statement that the induced potential (emf - electromotive force) in a magnetic field is directly proportional to the rate of change of flux is accurate, but it's also essential to understand how this principle applies to different scenarios.
In the case of a wire moving perpendicularly in a uniform magnetic field at a constant velocity, you're correct that there might not be an apparent change in the magnetic flux through the wire. However, the key factor here is the relative motion between the wire and the magnetic field.
When the wire is moving through the magnetic field, even at a constant velocity, the charges within the wire (usually electrons) experience a force due to their interaction with the magnetic field. This force causes a redistribution of charges within the wire, resulting in a buildup of charge on one end of the wire and a deficit of charge on the other end. This separation of charge creates an electric field within the wire.
This electric field, caused by the redistribution of charges, leads to the potential difference (voltage) across the ends of the wire. This potential difference is what we refer to as the induced emf. So, even though there might not be a change in the magnetic flux itself, the relative motion between the wire and the magnetic field leads to the redistribution of charges and the creation of an electric field, resulting in an induced potential difference.
In summary, the concept of electromagnetic induction is more subtle than just focusing on the rate of change of magnetic flux. It also involves the interaction between charges, electric fields, and magnetic fields, which can lead to the generation of an induced potential even in cases where the magnetic flux itself doesn't change significantly.