(a) A metal rod of length `L` rotates about an end with a uniform angular velocity `omega` . A uniform magnetic field `B` exists in the direction of the axis of rotation. Calculate the emf induced between the ends of the rod. Neglect the centripetal force acting on the free electrons as they move in circular paths.
(b) A rod of length `L` rotates with a small but uniform angular velocity `omega` about its perpendicular bisector. A uniform magnetic field `B` exists parallel to the axis of rotation. Find the potential difference (i) between the centre of rod and one end and (ii) between the two ends of the rod.

Explanation

Here the velocity of every part of rod is different. Taking an element of length `dx` at distance `x`
`de=Bvdx`
`eBomegaint_(0)^(L)xdx`
`=Bomega|(x^(2))/(2)|_(0)^(L)`
`e=(1)/(2)BomegaL^(2)` (Remember it)
`V_(0)-V_(A)=e=(1)/(2)BomegaL^(2)`
`V_(0)-V_(A)=(1)/(2)Bomega((L)/(2))^(2)=(BomegaL^(2))/(8)`
`V_(A)+e_(1)-e_(2)=V_(B)`
`V_(A)-V_(B)=0`