A uniform constant magnetic field `B` is directed at an angle of `45^(@)` to the `x axis` in the ` xy`- plane . ` PQRS` is a rigid, square wire frame carrying a steady current `I_(0)`, with its centre at the origin `O`. At time ` t = 0`, the frame is at rest in the position as shown in figure , with its sides parallel to the ` x and y` axis. Each side of the frame is of mass `M` and length `L`.
(a) What is the torque `tau` about `O` acting on the frame due to the magnetic field?
(b) Find the angle by which the frame rotates under the action of this torque in a short interval of time `Deltat`, and the axis about this rotation occurs .`( Deltat is so short that any variation in the torque during this interval may be neglected .) Given : the moment of interia of the frame about an axis through its centre perpendicular to its about an axis through its centre perpendicular to its plane is `(4)/(3) ML^(2)`.

Explanation

When the ring is not rotating `mg=2T_(0)`
When the ring is rotating its magnetic movement
`M=(Q)/(2pi)omegaxxpir^(2)`
Let the tension in the string be `T_(1)` and `T_(2)`
`T_(1)+T_(2)=mg,` Torque acting on the ring
`tau=(Qomega Br^(2))/(2)`
`T_(1)xx(D)/(2)-T_(2)xx(D)/(2)=(Q omegaBr^(2))/(2)impliesT_(1)=(mg)/(2)+(QomegaBr^(2))/(2D)`
maximum tension `=3T_(0)//2`
`(3T_(0))/(2)=T_(0)+(Qomega_(max)Br^(2))/(2D)impliesomega_(max)=(DT_(0))/(BQr^(2))`