com www.GOALias.blogspot.com

www.GOALias.blogspot.com158PhysicsFIGURE 4.22 (a) The area vector of the loopABCD makes an arbitrary angle θ withthe magnetic field. (b) Top view ofthe loop. The forces F 1and F 2actingon the arms AB and CDare indicated.the coil to be angle θ (The previous casecorresponds to θ = π/2). Figure 4.22 illustratesthis general case.The forces on the arms BC and DA are equal,opposite, and act along the axis of the coil, whichconnects the centres of mass of BC and DA. Beingcollinear along the axis they cancel each other,resulting in no net force or torque. The forces onarms AB and CD are F 1and F 2. They too are equaland opposite, with magnitude,F 1= F 2= I b BBut they are not collinear! This results in acouple as before. The torque is, however, less thanthe earlier case when plane of loop was along themagnetic field. This is because the perpendiculardistance between the forces of the couple hasdecreased. Figure 4.22(b) is a view of thearrangement from the AD end and it illustratesthese two forces constituting a couple. Themagnitude of the torque on the loop is,a aτ = F1 sinθ + F2sinθ2 2= I ab B sin θ= I A B sin θ (4.27)As θ à 0, the perpendicular distance betweenthe forces of the couple also approaches zero. Thismakes the forces collinear and the net force andtorque zero. The torques in Eqs. (4.26) and (4.27)can be expressed as vector product of the magnetic moment of the coiland the magnetic field. We define the magnetic moment of the currentloop as,m = I A (4.28)where the direction of the area vector A is given by the right-hand thumbrule and is directed into the plane of the paper in Fig. 4.21. Then as theangle between m and B is θ , Eqs. (4.26) and (4.27) can be expressed byone expressionτ = m × B(4.29)This is analogous to the electrostatic case (Electric dipole of dipolemoment p ein an electric field E).τ = p × EeAs is clear from Eq. (4.28), the dimensions of the magnetic moment are[A][L 2 ] and its unit is Am 2 .From Eq. (4.29), we see that the torque τ vanishes when m is eitherparallel or antiparallel to the magnetic field B. This indicates a state ofequilibrium as there is no torque on the coil (this also applies to anyobject with a magnetic moment m). When m and B are parallel the