Chapter 14 Magnetic Fields: VIl Magnetization M The equivalent surface Current densit The equivalent Volume Current Denisity J Calculation of Magnetic fields in materia Magnetic Field intensity H Amperes circuit Law Magnetic Susceptibility permeability a Magnetization curve ■ Hysterisis
Chapter 14 Magnetic Fields:VII ◼ Magnetization M ◼ The Equivalent Surface Current Density ◼ The Equivalent Volume Current Denisity J ◼ Calculation of Magnetic Fields in Material ◼ Magnetic Field Intensity H ◼ Ampere’s Circuit Law ◼ Magnetic Susceptibility, Permeability ◼ Magnetization Curve ◼ Hysterisis
14.1 Magnetization M So far we have studied only those magnetic fields due to free charges In fact all bodies contain spinning electrons moving in orbits, and they also produce magnetic fields Magnetized: Magnetic materials are similar to di- electrics, charges can possess magnetic moments, and these together can produce a resultant magnetic moment in a macroscopic body. Such a body is said to be magnetized
In most atoms the magnetic moments associated with the orbital and spinning motions of electrons cancel Paramagnetic: If the cancellation is not complete the material is said to paramagnetic When such a substance is put into a magnetic field gment→< E random motion Diamagnetic: in diamagnetic materials the elemen tary magnetic moments are not permanent, but are d induce
Ferromagnetic: in ferromagnetic materials such as iron the magnetization can be much larger than that of para- or diamagnetic substances Domain: in ferromagnetic materials, there are re gions in which electron spins are aligned. the mag netization of one domain may be oriented at random w.r.t. that of a neighboring domain Remark: usually dielectric DX E, linear; but in ferromagnetic materials the magnetization is non linear and depends on history
14.1 The Magnetization The magnetization M is the magnetization moment per unit volume at a given point If m is the average magnetic dipole moment per atom and they are aligned in the same direction then the magnetization M=Nm where N is the number density of atoms M has unit: ampere/meter
14.2 The Equivalent Surface Current density ae Recall: a loop of area tr and of current i has a magnetic moment m=TR I See Fig 14-1 .I meter M Figure 14-1 Ampere's model for the equivalent current in a cylinder of magnetized material
Each square cell has a cross-sectional area an a length l, and carries a surface current density a (ampere per meter) Each cell has a magnetic moment m= ala Each cell has a magnetization M=m/volume= a Inside all the currents cancel except at the bound ary with the surface current density a=M Thus, taking into account of the direction, one gets e M×n 1 called the equivalent surface current densi Recall the bound surface charge density Ob=Pn
14. 3 The equivalent Volume Current Density Je One molecule has a current with an area a, so its magnetic dipole moment is m= lae Consider a small length dl Taking dl as the hight and a cos o as the cross sectional area, we get a small volume adl cos o A molecule in this volume will contribute a current lad cos o all the molecules in this volume contribute a current Nadl cos0= Nm.dl=m. dl
Integrating dl over the path C yields the equivalent current le going through 亿M:dl=le, or in the differential form Y×M=Je ere Te=s Je da with S being the e surface boun y the closed pa Conclusion: the magnetization M gives rise to the equivalent current densities ae or Je
14.4 Magnetic Field Originating in Magnetized Ma terial In magnetized material the equivalent current den sities ae or Je also generate its magnetic field B just as if they were situated in a vacuum V×B=0Je
