- 1 -Chapter 6. Magnetostatic Fields in Matter6.1. MagnetizationAny macroscopic object consists of many atoms or molecules, each having electric chargesin motion. With each electron in an atom or molecule we can associate a tiny magnetic dipolemoment (due to its spin). Ordinarily, the individual dipoles cancel each other because of therandom orientation of their direction. However, when a magnetic field is applied, a netalignment of these magnetic dipoles occurs, and the material becomes magnetized. The state ofmagnetic polarization of a material is described by the parameter M which is called themagnetization of the material and is defined asM = magnetic dipole moment per unit volumeIn some material the magnetization is parallel to B . These materials are called paramagnetic. Inother materials the magnetization is opposite to B . These materials are called diamagnetic. Athird group of materials, also called Ferro magnetic materials, retain a substantial magnetizationindefinitely after the external field has been removed.ms1 q F Fq yza) zxFr Fl b) I s2Figure 6.1. Force on a rectangular current loop.- 2 -6.1.1. ParamagnetismConsider a rectangular current loop, with sides s1 and s2, located in a uniform magnetic field,pointing along the z axis. The magnetic dipole moment of the current loop makes an angle qwith the z axis (see Figure 6.1a). The magnetic forces on the left and right sides of the currentloop have the same magnitude but point in opposite directions (see Figure 6.1b). The net forceacting on the left and right side of the current loop is therefore equal to zero. The force on thetop and bottom part of the current loop (see Figure 6.1a) also have the same magnitude and pointin opposite directions. However since these forces are not collinear, the corresponding torque isnot equal to zero. The torque generated by magnetic forces acting on the top and the bottom ofthe current loop is equal toN = r ¥ F Â=s12F sinq ˆ i +s12F sinq ˆ i = s1Fsinq ˆ i The magnitude of the force F is equal toF = Idl ¥ B LineÚ= Is2BTherefore, the torque on the current loop is equal toNssIB imB i m B===¥12sin√sin√qq where m is the magnetic dipole moment of the current loop. As a result of the torque on thecurrent loop, it will rotate until its dipole moment is aligned with that of the external magneticfield.In atoms we can associate a dipole moment with each electron (spin). An external magneticfield will line up the dipole moment of the individual electrons (where not excluded by the Pauliprinciple). The induced magnetization is therefore parallel to the direction of the externalmagnetic field. It is this mechanism that is responsible for paramagnetism.In a uniform magnetic field the net force on any current loop is equal to zero:F = Idl ¥ B []Ú= Idl Ú[]¥ B = 0since the line integral of dl is equal to zero around any closed loop.If the magnetic field is non-uniform then, in general, there will be a net force on the currentloop. Consider an infinitesimal small current square of side e, located in the yz plane and with acurrent flowing in a counter-clockwise direction (see Figure 6.2). The force acting on the currentloop is the vector sum of the forces acting on each side:- 3 -F = Idl ¥ B Side 1Ú+ dl ¥ B Side 2Ú+ dl ¥ B Side 3Ú+ dl ¥ B Side 4ÚÈ Î Í Í ˘ ˚ ˙ ˙ == Iˆ j ¥B 0,y,0()()dy +ˆ k ¥ B 0,e,z()()dz0eÚ+ˆ j ¥ B 0,y,e()()dy +ˆ k ¥ B 0,0,z()()dze0Úe0Ú0eÚÈ Î Í ˘ ˚ ˙ == Iˆ j ¥B 0,y,0()- B 0,y,e()[]()dy +ˆ k ¥ B 0,e,z()- B 0,0,z()[]()dz0eÚ0eÚÈ Î Í ˘ ˚ ˙ == I -ˆ j ¥e∂B ∂z(0,y,0)Ê Ë Á ˆ ¯ ˜ dy +ˆ k ¥e∂B ∂y(0,0,z)Ê Ë Á ˆ ¯ ˜ dz0eÚ0eÚÈ Î Í Í ˘ ˚ ˙ ˙ 3yzxIe e 124Figure 6.2. Infinitesimal square current loop.In this derivation we have used a first-order Taylor expansion of B :B 0,e,z()= B 0,0,z()+e∂B ∂y(0,0,z)andB 0,y,e()= B 0,y,0()+e∂B ∂z(0,y,0)Assuming that the current loop is so small that the derivatives of B are constant over theboundaries of the loop we can evaluate the integrals and obtain for the total force:- 4 -F = Ie2ˆ k ¥∂B ∂y-ˆ j ¥∂B ∂zÈ Î Í ˘ ˚ ˙ = m∂Bx∂yˆ j -∂By∂yˆ i Ê Ë Á ˆ ¯ ˜ -∂Bz∂zˆ i -∂Bx∂zˆ k Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ = m∂Bx∂xˆ i +∂Bx∂yˆ j +∂Bx∂zˆ k È Î Í ˘ ˚ ˙ where m is the magnetic dipole moment of the current loop. In this derivation we have used thefact that the divergence of B is equal to zero for any magnetic field and this requires that∂Bx∂x=-∂By∂y-∂Bz∂zThe magnetic dipole moment m of the current loop is equal tom = mˆ i Therefore, the equation for the force acting on the current loop can be rewritten in terms of m :F =∂∂xmBx()ˆ i +∂∂ymBx()ˆ j +∂∂zmBx()ˆ k = — m ∑ B ()Any current loop can be build up of infinitesimal current loops and thereforeF = — m ∑ B ()for any current loop.Example: Problem 6.1.a) Calculate the torque exerted on the square loop shown in Figure 6.3 due to the circular loop(assume r is much larger than a or s).b) If the square loop is free to rotate, what will its equilibrium orientation be?a) The dipole moment of the current loop is equal tom =pa2I ˆ k where we have defined the z axis to be the direction of the dipole. The magnetic field at theposition of the square loop, assuming that r»a, will be a dipole field with q = 90°:B =m04pmr3ˆ q =m04ppa2Ir3ˆ q =-m04a2r3I ˆ k The dipole moment of the square loop is equal tom square= s2I ˆ i- 5 -Ima s IrFigure 6.3. Problem 6.1.Here we have assumed that the x axis coincides with the line connecting the center of the currentcircle and the center of the current square. The torque on the square loop is equal toN = m square¥ B =ˆ i ˆ j ˆ k s2I 0000-m04a2r3I=m04a2s2r3I2ˆ j b) Suppose the dipole moment of the square loop is equal tom square= mxˆ i + myˆ j + mzˆ k The torque on this dipole is equal toN = m square¥ B =ˆ i ˆ j ˆ k mxmymz00-m04a2r3I=-m04a2r3I my ˆ i +m04a2r3I mx ˆ j In the equilibrium position, the torque on the current loop must be equal to zero. This thereforerequires thatmx= my= 0Thus, in the equilibrium position the dipole will have its dipole moment directed along the z axis.The energy of a magnetic dipole in a magnetic field is equal to- 6 -U =-m ∑ B The system will minimize its energy if the dipole moment and the magnetic field are parallel.Since the magnetic field at the position of