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

Answer

a

Explanation

Let a is the side of squre.

Torque of current =`MB sin (90- theta)`
`=MBcos theta =ia^2B cos theta`
Mass of each side =`m=rhoAa`
Torque of gravity `=2mga/2 sin theta+mga sin theta`
`=2mga sin theta=2rhoAga^2sin theta`
Now, both torques should be same
i.e., `ia^2B cos theta=2rhoAga^2 sin theta`
`implies cot theta=(2rhoAg)/(iB)`