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. `(2Arhog)/(iB)=cottheta`
B. `(2Arhog)/(iB)=tantheta`
C. `(Arhog)/(iB)=sintheta`
D. `(Arhog)/(2iB)=costheta`

Answer

A

Explanation

Let `a` is the side of square.
Torque of current `=MB sin (90-theta)`
`=Mbcostheta=ia^(2)Bcostheta`

Mass of each side `=m=rhoAa`
Torque of gravity `= 2mg(a)/(2)sintheta+mg a sintheta`
`=2mga asintheta=2rhoAga^(2)sintheta`
Now both torques should be same
` Ia^(2)Bcostheta=2rhoAga^(2)sintheta`
`implies cottheta=(2rhoAg)/(IB)`