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

(a)from `P` to `Q`
(b)`IbB_(c)(3hatk-4hati)`
( c) `(mg)/(6bB_(0))`

Explanation

(a) and (c )
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 `vecM=-lab hatk`
Torque on the loop due to magnetic force is:
`vectau_(1)=vecMxxvecB`
`=(-lab hatk)xx{(3 hati+4 hat k)B_(0)}`
`=-3labB_(0) hatj`
Torque of weight of the loop about axis `PQ` is
`vectau_(2)=vecrxxvecF`
`=(a/2 hati)xx(-mg hatk)`
`=(mga)/2 hatj`
We see that when the current in the wire `PQ` is from `P` to `Q` ,`tau_(1)` and `tau_(2)` are in opposite direction so they can cancel each other and the loop may remain in equilibrium.So the direction of current `l` in the wire `PQ` is from `P` to `Q`. Further for equilibrium of the loop.
`|vectau_(1)|=|vectau_(2)|`
or `3labB=(mga)/2`
`l=(mg)/(6bB_(0))`
(b)Magnetic force on wire `RS` is:
`F=I(vecexxB)`
`=l[(-b hatj)xx{(3 hatj)+4hatk)B_(0)}]`
or `vecF=IbB_(0)(3hatk-4hatj)`