A wire of cross-sectional area A forms three sides of a square and is free to rotate about axis OO'. If the structure is deflected by an angle `theta` from the vertical when current i is passed through it in a magnetic field B acting vertically upward and density of the wire is `rho`, then the value of `theta` is given by

Options

A. `sin theta=(Arhog)/(IB)`
B. `cos theta=(ArhoB)/(2IB)`
C. `tan theta=(2Arhog)/(IB)`
D. `cot theta=(2Arhog)/(IB)`

Explanation

Mass of each side of square, `m=Aarho`
Magnetic moment of current loop, `M=Ia^2`,
acting perpendicular to the plane of current loop. In the deflected position of current loop, figure, the perpendicular distance of C or D from axis of rotation `=a/2sin theta`.
Perpendicular distance of weight mg of wire EF from axis of rotation `CD=a sin theta`
Torque on current square loop in magnetic field
`=MB sin (90^@-theta)=MB sin theta`
`=Ia^2B cos theta`

Torque on loop due to gravity
`=2mg(a//2)sin theta+mg a sin theta`
`=2mg a sin theta=2A a rho g xx a sin theta`
`=2 A a^2 rho g sin theta`
In deflected position, the above two torques are equal and opposite, so
`Ia^2Bcostheta=2Aa^2rhogsin theta`
or `cot theta=(2Arhog)/(IB)`