A copper wire formed into a semicircle of radius r translates in its plane at a constant velocity v . A uniform magnetic field B is directed perpendicular to the wire's plane. Determine the emf induced between the ends of the wire given that the velocity is perpendicular to the diameter connecting the free ends.

Step-by-step solution

The motional emf induced in a conductor moving in a uniform magnetic field depends only on its effective length, which is the straight line distance between its free ends. For a semicircular wire of radius r , the effective length vector L is the diameter connecting its ends, so | L | = 2r . The induced emf is given by = ( v B ) L . Since the magnetic field B is perpendicular to the plane of the wire and the velocity v is perpendicular to the diameter, the vectors v , B , and L are mutually perpendicular. Thus, the