A rectangular wire farme can be rotated through it arm PQ which is hinged along the y-axis as shown the figure . If the wire-frame has mass and externt magentic field is `B_(0)(hati+hatj)`. The magnetic field at due to the current in the wire PQ will be have the direction (assume the wire-fram is the in horizontal equilibrium positive )

Answer

(i) 0
(ii) 6

Explanation

(i) 0 (ii) 6
(i)Let the direction of current in wire PQ is from P to Q and its magnitude be I.
The magnetic moment of the given loop is
`M=-I abhatk`
Torque on the loop due to magnetic force is
`tau_(1)=MxB=(-Iabhatk)xx(3hati+4hatk)B_(0)hati=-3IabB_(0)hatj`
Torque of weight of the loop about axis PQ is
`tau_(2)=rxxF=(a/2hati)xx(-mghatk)=(mga)/2hatj`
We see that when the current in the wire PQ is from P to Q `tau_(1)` and `tau_(2)` are in opposite directions, so they can cancel each other and the loop may remain in equilibrium. So, the direction of current I in wire PQ is from P to Q. Further for equilibrium of the loop
`|tau_(1)|=|tau_(2)|` or `3IabB_(0)=(mga)/2:I=(mg)/(6bB_(0))`

(ii) Magnetic force on wire RS is
`F=I(IxxB)=K[(-bhatj)xx{(3hati+4hatk)B_(0)}], F=IoB_(0)(3hatk-4hati)`