In a trapeze-shaped structure, two rigid wires of negligble mass support a conducting bar of mass m and length L as shown in Fig. A source of emf is applied to the wires so that a current I flows through the bar. A uniform magnetic field `vec B` is perpendicular to the plane of the wires and bar.
a. Compute the current that the source of emf must provide so that there is no tension in the wires.
b. If the current is reduced to half the value computed in (a) and the plane of the structure is moved through an angle `theta`, compute the tension in the wires and the magnitude of the net unbalanced force on the bar at the instant it is released from this angle.

Answer

(a) `i=(mg)/(LB)` ; (b) `T=(mg)/4 cos theta; F_("net")=(mg)/2 cos theta`

Explanation

(a) Writing torque equation about A
`vectau=(mg-F_B)L/2-TL`
Since the rod is in equilibrium, there is no torque in vertical
directio.
`implies TL=1/2(mg-F_B)`
For wires to be tension free, `T=0`
`implies F_B=mg implies iLB implies mg implies i=(mg)/(LB)`

(b) If the wire is in equilibrium, the forces acting on it are as shown in Fig.
`i=(mg)/(2LB) :. F_B=1/2mg`
For tension in the wire, `2T+F_B cos`
`theta=mg cos theta`
`T=(mg)/4 cos theta`
Net unbalanced force on the rod will be in the direction perpendicular to the wire

`F_(n et)=mgsin theta-F_B sin theta`
`F_(n et) =(mg)/2 cos theta`
.