A wire loop enclosing as semicircle of radius `R` is located on the boudary of uniform magnetic field `B`. At the moment `t=0`, the loop is set into rotation with a costant angular acceleration `alpha` about an axis `O` coinciding with a line of vector B on the boundary. Find the emf induced in the loop as a function of time. Draw the approximate plot of this function.The arrow in the figure shows the emf direction taken to be positive.

Explanation

Flux at any moment of time,
`|Phi_(t)| = |vec(B). D vec(S)| = B ((1)/(2) R^(2) varphi)`
where `varphi` is the sector angle, enclosed by the field Now, magnitude of induced `e.m.f` is given by,
`xi_("in") = |(d Phi_(t))/(dt)| = |(B R^(2))/(2) (d varphi)/(dt)| = (BR^(2))/(2) omega`,
where `omega` is the angular velocity of the disc. But as it series rotating from rest at `t = 0` with an angular accelearation `beta` its in angular velocity `omega (t) = beta t`. So,
`xi_(i n) = (B R^(2))/(2) beta t`.
According to Lenz law the first half cycle current in the loop is in anticlockwise sense, and in subsequent half cycle it is in clockwise sense.
Thus is genral, `xi_(i n) = (-1)^(n) (B R^(2))/(2) beta t`, where `n` in number of half revolutions.
The plot `xi_(i n) (t)`, where `t_(n) = sqrt(2 pi//beta)` is shown in the answer sheet.