A conducting loop of area `A` is placed in a magnetic field which varies sinusoidally with time as `B=B_(0)sin`

Explanation

(a) `phi=Bacostheta=B_(0)sinomegatcos60^(@)=(B^(@)Asinomegat)/(2)`
`e=(dphi)/(dt)=(B_(0)Aomegacosomegat)/(2)`
`e` is maximum ,when `cosomegat=I`
`e_(max)=(I)/(2)B_(0)Aomega`
(b) `e=(B_(0)Aomega)/(2)cosomegat`
At `t=T//6`
`e=(B_(0)Aomega)/(2)cos((omegaT)/(6))`
`=(B_(0)Aomega)/(2)cos((pi)/(3))=(B_(0)Aomega)/(4)( :. omega=(2pi)/(T))`
`I=e//R=(B_(0)Aomega cos omega t)/(2R)`
`q=int_(0)^(T)Idt=(B_(0)Aomega)/(2R)int_(0)^(T)cosomegatdt=0`
(d) `H=int_(0)^(T)(e^(2))/(R)dt=(B_(0)^(2)A^(2)omega^(@))/(4R)int_(0)^(T)cos^(2)omegatdt`
`=(B_(0)^(2)A^(2)omega^(2))/(4R)(T)/(2)=(B_(0)^(2)A^(2)omega^(2))/(4R)(pi)/(omega)`
`=(piB_(0)^(2)A^(2)omega)/(4R)`