A rod of mass `m` can rotate without friction about axis sliding (also without friction) along a conducting ring of radius `b` arranged in a vertical plane as shown in the . The entire arrengement is placed in a uniform magnetic field with the inducetion `B` perpendicular to the plane of the ring. The axis and the conductor are connected to teh terminals of a current source. Determine
(a) according to which law current `I` flowing in the rod must vary for the rod to rotate at a constant angular speed. Being to measure the time from the instant when the rod is in its right-hand horizontal position. Consider the current to be positive when it flows from the axis of rotation toward the ring.
(b) what emf `E` of the source must be applied to maintain the required current? Consider the total resistance of the circuit to be constant and equal to `R`. Disergard the inductance of teh circuit.

Answer

(a) `(mg cos omegat)/(Bb)`; (b) `(Bomegab^(2))/(2) + (mg R cos omega t)/(Bb)`

Explanation

(a) `tau_(B) = F_(B) xx (b)/(2) rarr tau_(B) = IbB xx (b)/(2) = IB(b^(2))/(2)`
`tau_(net) = 0 `rarr` mg cos theta xx (b)/(2) = (IBb^(2))/(2)`
`rarr` `mg cos theta = IBb`
`theta = omegat`
`rarr` `I = (mg cos omegat)/(Bb)`

(b) `((E - E_(i nd)))/(R) = I`, `E = E_(i nd). + IR`
`rarr` `E = (Bomegab^(2))/(2) + (mg R cos omegat)/(Bb)`