(a) Derive the expression for the torque acting on a current carrying placed in a magnetic field.
(b) Explain the significance of a radial magnetic when current carrying coil is kept in it.

Explanation

Let us consider a rectangular coil ABCD carrying I, Placed in a magnetic field B making an angle `theta` with the plane of the coid and `phi` with the area vector of the plane.
`vec(F_(b))=vec(-F_(b))`
`vec(F_(I))=vec(-F_(I))`
So, they cancel out in pairs but `vecF_(l)" and "vec(-F_(I))` provide torque to the ractangular coil ABCD.
`tau="(Force )(Arm of Couple)"`
`tau="Force"xx"Perpendicular distance between two action lines of forces"`
`tau=I l B xx b cos theta`
`tau=IlB b sin phi`
let the number of turns be N
`{:(tau="NIA B sin "phi,"{l" xx b=A),(tau="MB sin "phi, "NIA = M"):}" "{{:(theta+phi=90^(@)),(theta=90^(@)-phi),(costheta=sinphi):}`
Magnetic diploe moment
`vec(tau)=vec(M)xxvec(B)`

Significance of radical magnetic field : When apply radical field as shown in the figure `phi` is always equals to `90^(@)`.
`tau=MB sin 90^(@)`
`sin 90^(@)=1`
Hence, Torque maximum and constant.