A metal rod of length $L$ rotates with angular velocity $\omega$ about an axis passing through one end and perpendicular to the rod in a uniform magnetic field $B$ directed parallel to the axis of rotation. What is the induced emf across the ends of the rod?

Solution

1. The motional emf induced in a conductor is given by the integral $\int (\vec{v}\times \vec{B})\cdot d\vec{l}$.
2. The magnetic field $\vec{B}$ is directed parallel to the axis of rotation, while the velocity $\vec{v}$ of any segment of the rod lies in the plane perpendicular to the axis.
3. The cross product $\vec{v}\times \vec{B}$ results in a vector that is perpendicular to the rod, meaning it has no component along the length element $d\vec{l}$ of the rod.
4. Since the dot product $(\vec{v}\times \vec{B})\cdot d\vec{l}$ is zero at every point along the rod, the total induced emf is zero.
5. Hence the answer is (C).