Electromagnetism Zhu Jiongming Department of Physics Shanghai Teachers University

Electromagnetism Chapter 1 Electric Field Chapter 2 Conductors Chapter 3 Dielectrics Chapter 4 Direct-Current Circuits Chapter 5 Magnetic Field Chapter 6 Electromagnetic Induction Chapter 7 Magnetic MaterialsMagnetic Materials Chapter 8 Alternating Current Chapter 9 Electromagnetic Waves

Chapter 7 Magnetic Materials §1. Basic Laws in Magnetic MaterialsBasic Laws in Magnetic Materials §2. Paramagnetism and DiamagnetismParamagnetism and Diamagnetism §3. Ferromagnetism and FerromagneticsFerromagnetism and Ferromagnetics §4. Magnetic Field EnergyMagnetic Field Energy

§1. Basic Laws in Magnetic Materials 1. Magnetizing and Magnetization of MaterialsMagnetizing and Magnetization of Materials ● Magnetizing ● Magnetization 2. Magnetization CurrentMagnetization Current 3. Magnetic Field Intensity Ampere’s Law in Magnetic Materials 4. ExamplesExamples

1. Magnetizing and Magnetization magnetic material  magnetized  loop model for electron orbits Comparison ： dielectric material  polarized  electric dipole F MagnetizingMagnetizing F MagnetizationMagnetization

Magnetizing of Materials loop model  atomic magnetic dipole moment p m  aligned （ magnetized ） i pmpm I free current （ conducting ） I 0  B 0 bound current （ magnetized ） I’  B’ B = B 0 + B’  cancel out inside  bound current on outline

Magnetization F Magnetization ： vector sum of p m per unit volume F Uniform ： M is same at every point in material compare ： F Experiments ： in isotropic non-ferromagnetics M = gB compare ： P =  0 E  paramagnetics ： g > 0 M and B same direction  diamagnetics ： g < 0 M and B opposite F magnetic materials paramagnetic diamagnetic ferromagnetic non-ferromagnetic

2. Magnetization Current （ 1 ） surface S ， border L bound current I’ passing through S L S Only the loop currents round L contribute to I’

2. Magnetization Current （ 1 ） L S surface S ， border L bound current I’ passing through S Only the loop currents round L contribute to I’

 volume ： Sdl cos   number of loop ： NSdl cos  （ center of which in the cylinder ）  current ： dI’ = I m NSdl cos  2. Magnetization Current （ 2 ） N ： number of molecules per unit volume I m ： molecule’s loop current Compare ：  M dldl S inclined cylinder ： length dl, base area S of loop

2. Magnetization Current （ 3 ） Surface current density —— charges perpendicularly passing through unit length per unit time ( on surface ) dldl dI’dI’ F It is proved （ skip ）  Uniformly magnetized ： j’ = 0  Interface between two materials ：  ’ = ( M 2 - M 1 )  n （ n ： 2  1 ） Compare ： uniform  ’ = 0  ’ = ( P 2 - P 1 ) · n

3. Ampere’s Law in Magnetic Materials Vacuum ： M =0 ， B =  0 H Definition ： D   0 E + P Electric displacement Magnetic field intensity

Magnetic Field Intensity H D   0 E + P

Magnetic Field Intensity H Magnetic Field Intensity H Relationship between B and H B =  0 ( H + M ) M =  m H  B =  0 ( H +  m H ) Magnetic susceptibility ：  m Relative permeability ：  r = 1 +  m Permeability ：  =  0  r Permeability of vacuum ：  0

Example 1 （ p.288 / [Ex.] ） Toroidal solenoid ， r << R Known ： I 0, n, , V Find ： H, B, L Sol. ： inside ： take a loop as shown R r （ outside ： H out = 0, B out = 0 ）  S = B in S =  nI 0 S ，  S = N  S = n2  R·  nI 0 S =  n 2 V I 0  L =  S / I 0 =  n 2 V （ vacuum ： L 0 =  0 n 2 V ） =  r L 0

Example 2 （ p.315 / ）（ 1 ） Uniformly magnetized sphere M R  z o Known ： R, M （ along z axis ） Find ：  ’ （ on surface ）， p m Sol. ： Material 2 ， vacuum 1 n = r ( 2  1 ) M 2 = M, M 1 = 0 （ direction ： right hand rule ）

Example 2 （ p.315 / ）（ 2 ） （ direction ： along z ） Consider ： a vacuum sphere in an infinite magnetic material （ find  ’ on surface etc. ）

Exercises p.315 / , 3, 4

§2. Paramagnetism and Diamagnetism 1. ParamagnetismParamagnetism 2. DiamagnetismDiamagnetism

1. Paramagnetism Molecules, atoms  electrons orbital / spin magnetic dipole moment  not cancel ： permanent magnetic moment p m  cancel out ： p m is 0 ( of a molecule ) p m  paramagnetism  external field B = 0 ， thermal motion  p m orient randomly ， cancel out  M = 0  B  0 ， p m tend to line up with B  M  0 paramagnetic material ： p m  0 include ： aluminium, natrium, oxygen etc.

2. Diamagnetism （ 1 ） Exhibited by all materials, very weak, opposite to B  material with p m  0 ， exhibit paramagnetism  material with p m = 0 ， exhibit diamagnetism electron moving in a circular path ， B introduced ， r no change,  increased  p m changed, opposite to B  diamagnetism 00 p m0 -e-e v

2. Diamagnetism （ 2 ） Assume B and  0 in the same direction B -e-e F Lorentz force ： F L = e ( v 0 +  v ) B Centripetal force ： EIEI

2. Diamagnetism （ 3 ）  -e-e  r unchanged ，  > 0   p m opposite to B  diamagnetism, very weak pmpm B

§3. Ferromagnetism 1. Magnetizing Properties of Ferromagnetic MaterialMagnetizing Properties of Ferromagnetic Material 2. Classification and ApplicationClassification and Application 3. Magnetic DomainsMagnetic Domains non-ferromagnetism ： M = gB =  m H B =  H 0 H B ferromagnetism ：？ —— experiment

Experiment A R 1 2 test B Experiment ： Adjust R to control I obtain H = nI Electromagnetic induction SKIP EXPERIMENT

F continue ， B changing slow down F beyond S ， H  ， B almost not change （ saturation magnetic intensity H S ）  H  to 0 ， B - H curve not retraced F H = 0 ， B  0 （ residual field ） magnetic hysteresis  switch to 2 ， I reversed ， H ： 0  - H S  H = - H D ， B = 0 ， H D rectification force  H  ： - H S  H S ， closed ， hysteresis loop 1. Magnetizing Properties （ 1 ） F open switch ， H = 0 ， B = 0 ， O F R maximum ， switch to 1 ， A F R  ， H  ， B  ， AC A S C R o H B

1. Magnetizing Properties （ 2 ） F Saturation magnetic intensity H S F Residual field B R F Magnetic hysteresis F Rectification force H D F Hysteresis loop （ Symmetry about O ） F Small hysteresis loop o H B R S D S’ D’ R’ F H corresponding to many B Magnetization curve H and B ： 1 to 1

1. Magnetizing Properties （ 3 ） F Magnetization curve H and B ： 1 to 1 F Permeability of a ferromagnetic material ： O H B,  B   not a constant  very large  r ~ 10 4

2. Classification and Application F Three characteristics ：  high  ： strong field by weak current ， motors, transformers  Non-linearity ： non-linear elements  Magnetic hysteresis ： permanent magnets F Classification ：  Soft magnetism  Hard magnetism o H B Hard Soft

F Effect of external magnetic field ：  growth in size of the domains oriented along Bgrowth in size of the domains oriented along B  shift of the orientation of dipoles in a domain Magnetizing ： H  ， M  ， B  B =  0 (H + M ) saturation ： M stop increasing F Not reversible ： as H removed F Temperature ： thermal motion 3. Magnetic Domains （ Quantum theory ） F Magnetic domains ： magnetized regions H = 0 ， M = 0 T > T C paramagnetism T < T C ferromagnetism T C ： critical temperature Curie Point

Exercises p.316 /

§4. Magnetic Field Energy Energy density w m at every point in the magnetic field Consider a solenoid inside ： H = nI ， B =  nI （ uniform field ） outside ： H = 0 ， B = 0 Self-inductance ： Magnetic energy ： Energy density ： Non-uniform ： w m ( x, y, z )

Example A long coaxial cable made of cylinders of radius r 1 < r 2 and material of permeability  carries a current I. Find magnetic energy and self-inductance of a length l. Sol. ： H = I / 2  r ( r 1 < r <r 2 ) ， H = 0 ( r r 2 )

Exercises p.316 / , 2, 3