Shows a square frame of wire having a total resistance r placed coplanarly with a long, straight wire. The wire carries a current I given by `i= i_0 sin omega t`. Find(a) the flux of the magnetic field through the square frame, (b) the emf induced in the frame and (c) the heat developd in the frame in the time interval 0 to `(20pi)/(omega)`.

Explanation

Magnetic field due to long wire at distance `x`
`B=(mu_(0)i)/(2pix)`
Taking a small area of length a and width `dx`
`dA=adx`
`dphi=BdA=(mu_(0)ia)/(2pi)(dx)/(x)`
Total flux through the loop
`phi=(mu_(0)ia)/(2pi)int_(b)^(b+a)(dx)/(x)=(mu_(0)ia)/(2pi)|1nx|_(b)^(b+a)`
`=(mu_(0)ia)/(2pi)[1n(b+a)-1n(b)]`
`=(mu_(0)ia)/(2pi)1n(1+(a)/(b))`
`=mu_(0)i_(0)a1n((1+(a)/(b))omega)/(2pi)cosomegat`
(d) Induced emf
`e=(dphi)/(dt)=(mu_(0)ia1n(1+(b)/(a))omega)/(2pi)cosomegat`
(c) Heat developed in
`t=0` to `t=(40pi)/(omega)=20T`
`H=int_(0)^(20T)(e^(2))/(R)dt`
`=([mu(0)i_(0)aomega1n(1+(b)/(a))]^(2))/(4pi^(2)R)int_(0)^(20T)cos^(2)omegatdt`
`=[mu(0)i_(0)aomega1n(1+(b)/(a))
]^(2)/(4pi^(2)R)((T)/(2)xx20)`
`=[[mu(0)i_(0)aomega1n(1+(b)/(a))]^(2))/(4pi^(2)R).10.(2pi)/(omega)`
`=5[[mu_(0)i_(0)a1n((1+(b))/(a))]^(2)omega)/(piR)`