A rod of length `2a` is free to rotate in a vertical plane, about a horizontal axis `O` passing through its mid-point. A long straight, horizontal wire is in the same plane and is carrying a constant current i as shown in figure. At initial moment of time, the rod is horizontal and starts to rotate with constant angular velocity `omega`, calculate emf induced in the rod as a function of time.

Answer

A, B

Explanation

`theta=omegat`

`de=B(omegax)dx`
Here `B=mu_0/(2pi) i/(d-xsinomegat)`
`:. de=(mu_0iomega)/(2pi) x/(d-xsinomegat) dx`
`V_(OA) = V_(0) -V_(A)= int_(0)^(a) de=(mu_(o)i omega)/(2pi) int_(0)^(a) (x)/(d-x sin omegat) dx`
`=- (mu_(0)i omega)/(2pi sin omegat) [(d)/(sin omegat) In ((d- a sin omegat)/(d)) +a]`
Similarly,
`V_(OB) = V_(O) - V_(B) = (mu_(0)i omega)/(2pi) int_(0)^(a) (x)/(d + x sin omegat) dx`
`= (mu_(0) i omega)/(2pi sin omegat) [a-(d)/(sin omegat) In ((d+a sin omegat)/(d))]`
`:. V_(AB) = V_(OB) - V_(OA)`
`= (mu_(0)i omega)/(2pi sin omegat) [2a +(d)/(sin omegat) In ((d-a sin omegat)/(d +a sin omega t))]`