Question

A square loop of edge a having `N` turns is rotated with a uniform angular velocity `omega` about one of its diagonals which is kept fixed in a horizontal position. A uniform magnetic field `B` exists in the vertical direction. Find
(a) the emf induced ion the coil as a function of time `t`
(b) the maximum emf induced
(c) the average emf induced in the loop over a long period (d) the average of the squares of emf induced over a long period
(e) if resistance of loop is `R` , amount of change flow in time `t=0` to `t=2T`
(f) heat produced in time `t =0` to `=4T`

Answer 1

Initially plane of loop is `_|_^(ar)` to the field. Let at any time `t` , normal to loop makes an angle`theta` with magnetic field `theta=omegat` .
The flux passing through loop`phi=NBAcostheta=Nba^(2)cosomegat`
`e=-(dphi)/(dt)=Nba^(2)omegasinomegat=e_(0)sinomegat`
where `e_(0)=NBa^(2)omega`
(b) `e` is maximum, when `sinomegat=I`
`e_(max)=e_(0)`
(c) Long period i.e., one time period
`bar(e)=(int_(0)^(T)edt)/(int_(0)^(T)dt)=(I)/(T).e_(0)int_(0)^(T)sinomegatdt=0`
(d) `e^(2)=(int_(0)^(T)e^(2)dt)/(int_(0)^(T)dt)=e_(0)^(2)(1)/(T)int_(0)^(T)sin^(2)omegatdt=(e_(0)^(2))/(2)`
(e) Induced current `i=(e)/(R)=(e^(0))/(R)sinomegati_(0)=(e_(0))/(R)`
`q=int_(0)^(2T)i dt=i_(0)int_(0)^(2T)sinomegadt=0`
(f) `H=int_(0)^(4T)(e^(2))/(R)dt=(e_(0)^(2))/(R)int_(0)^(4T)sin^(2)omegadt`
`=(e_(0)^(2))/(R).4.(T)/(2)=(e_(0)^(2))/(R).2.(2pi)/(omega)`
`=(4pie_(0)^(2))/(omegaR)`