Sol. (i) When external magnetic field is absent, Since, rod is made of a metal, therefore, if has free electrons. When is rotated, free electrons start to move radially outwards. Due to this, at ends rod becomes negatively charged and at O it becomes positively charge. Hence, a radial, electric field is established in the rod which exerts a radially inward force on electron.

Dynamic equilibrium is established when this radially inward force exerted by electric field becomes equal to centripetal force required for circular motion of electrons.

Consider a point P at distance x from axis of rotation O as shown in Fig.

Let electric field strength of this point be E. For dynamic equilibrium of a free electron at P,

mx ω 2 = eE or E =

But E = –

∴ dV = – . xdx .....(1)

Let at end A, potential be V A and that at B be V B , Integrating equation (1),

or V B – V A = –

or V A – V B =

(ii) When a uniform magnetic field B exists, When rod rotates, it cuts magnetic flux, therefore, an emf is induced in the rod. Due to this emf a potential difference is established between its two ends.

Considering an elemental length dx of rod at a distance x from axis of rotation as shown in Fig. Velocity of this elemental length is

v = x ω

∴ EMF induced in elemental length dx is dV = Bvdx = B ω x dx

Integrating above equation,

V A – V B =