A metal rod of mass `m` can rotate about a horizontal axis `O`, sliding along a circular conductor of radius `a` (fig). The arrangment is located in a unifrom magnetic field of induction `B` directed perpendicular to the ring plane. The axis and the ring are connected to an emf source to form a circuit of resistance `R`. Neglecting the friction, circuit induction and ring resistance, find the law according to which the source and must very to make the rod rotate with a constant angular velocity `omega`.

Explanation

As the rod rotates, an emf.
`(d)/(dt) (1)/(2) a^(2) theta. B = (1)/(2) a^(2) B omega`
is induced in it. The net current in the conductor is then `(xi(t) - (1)/(2) a^(2) B omega)/(R)`
A magnetic force will then act on the conductor of magnitude `BI` per unit length. Its direction will be normal to `B` and the rod and its torque wil be
`int_(0)^(a) ((xi(t) - (1)/(2) a^(2) B omega)/(R)) dx B x`
Obviously both magnetic and mehanical torque acting on the `C.M.` of the rod must be equal but opposite in sense. Then
for equilibrium at constant `omega`
`(xi (t) - (1)/(2) a^(2) B omega)/(R) . (B a^(2))/(2) = (1)/(2) mga sin omega t`
or, `xi(t) = (1)/(2) a^(2) B omega + (mg R)/(aB) sin omega t = (1)/(2 aB) (a^(3) B^(2) omega + 2 mg R sin omega t)`
(The anser given in the book in incorrect dimensionally.)