A uniform conducting ring of mass `m=2kg`, radius `r=2m` and resistance `8Omega` is kept on smooth, horizontal surface. A time varying magnetic field `vecB=(hati+t^(2)hatj)` Tesla is present in the region, where `t` is time in second and take vertical as `y`-axis. (Take `pi^(2)=10`). Then

Options

A. Time when ring starts toppling is 1 sec
B. Time when ring starts toppling is `3//4` sec
C. Heat generated through the ring till the instant when the ring start toppling is `40//3` Joule.
D. Heat generated through the ring till the instant when the ring start toppling is `80//3` Joule

Answer

A, D

Explanation

`phi=BA=(t^(2))(pir^(2))`
`epsilon=(dphi)/(dt)=2pir^(2)t`
`i=(epsilon)/R=(2pir^(2)t)/R`
For toppling
`tau=|vec(mu)xxvecB|gemgr`
`(i.pir^(2)).1gemgr`
`(2pir^(2)t)/R.pir^(2) gemgr`
`t ge (mgR)/(2pi^(2)r^(3))=1`sec
Heat `=int i^(2)Rdt=int((2pir^(2)t)/R)^(2)=dt.R`
Heat `=(4pi^(2)r^(4))/R int_(0)^(1) t^(2) dt =80/3` Joule