A copper wire bent into a semicircle of radius r translates within its plane at a constant velocity v . A uniform magnetic field B exists perpendicular to the plane of the wire. Calculate the emf induced between the ends of the wire if the velocity is parallel to the diameter joining the free ends.

Step-by-step solution

The induced emf in a moving conductor in a uniform magnetic field is given by = ( v B ) l , where l is the effective length vector joining the ends of the conductor. For the given semicircular wire, the effective length vector l lies along the diameter, so its magnitude is 2r . The velocity v of the wire is given to be parallel to the diameter, which means v is parallel to l . Since v and l are parallel, the vector v B is perpendicular to l . Thus, the dot product ( v B ) l = 0 . Therefore, the induced emf between