Lecture 12 Magnetism of Matter: Maxwell’s Equations Chp. 32• Cartoon• Warm-up problem• Opening Demo• Topics– Finish up Mutual inductance– Ferromagnetism– Maxwell equations– Displacement current• DemosWhat is the magnetic energy stored in a solenoid orcoil dUBdt= Lididt dUB= Lidi dUB0UB!= Lidi0i! UB= Lidi0i!=12Li2 UB=12Li2For an inductor LNow define the energy per unit volume uB=UBAlArea Al uB=12Li2Al=Lli22A Ll=µ0n2A uB=12Li2Al=12µ0n2i2 uB=B22µ0 B =µ0ni uE=E22!0The energy density formulais valid in generalWhat is Mutual Inductance? MWhen two circuits are near one another and both have currentschanging, they can induce emfs in each other.On circuit boards you have to be careful you do not put circuits neareach other that have large mutual inductance.They have to be oriented carefully and even shielded.221111IMILm+=!112222IMILm+=!MMM ==21121 2I1I275. A rectangular loop of N closelypacked turns is positioned near along, straight wire as shown in thefigure.(a) What is the mutual inductance Mfor the loop-wire combination?(b) Evaluate M for N = 100, a = 1.0 cm,b = 8.0 cm, and l = 30 cm. ! = Bwireldr =µ0il2"rdr =aa +b#aa +b#µ0il2"ln raa +b=µ0il2"ln(1+ba)(a) The flux over the loop cross section due to the current i in the wire isgiven by M =N!i=Nµ0l2"ln 1 +ba# $ % & ' ( M =N!i(b) Evaluate M for N = 100, a = 1.0 cm,b = 8.0 cm, and l = 30 cm.(b) From the formula for M obtained,( )( )!"#$%&+'(=)0.10.81ln230.01041007**mmHMH5103.1!"= M =Nµ0l2!ln 1+ba" # $ % & 'FerromagnetismIron, cobalt, nickel, and rare earth alloys exhibit ferromagnetism.The so called exchange coupling causes electron magnetic momentsof one atom to align with electrons of other atoms. This alignment producesmagnetism. Whole groups of atoms align and form domains.(See Figure 32-12 on page 756)A material becomes a magnet when the domains line up adding all themagnetic moments.You can actually hear the domains shifting by bringing up an magnet and hear the induced currents in the coil. Barkhausen EffectTwo other types of magnetic behavior are paramagnetism or diamagnetism.What is the atomic origin of magnetism?Electron spinning on its axisElectron orbiting around the nucleusSpin Magnetic Dipole Moment of the Electron  ! µ = !em! S e = 1.6 "10!9Coulombsm = 9.11 "10!31kgS is the angular momentum due to the electron’s spin. It has units kg.m2/s. µ has units of A.m2 - current times areaRecall for a current loop, the magnetic dipole moment = current times area of loopIn the quantum field theory of the electron, S can not be measured. Only it’s component along the z axis can be measured. In quantumphysics, there are only two values of the z component of the electron spin.Therefore, only the z component of µ can bemeasured.Its two possible values are: µz= ±eh4!mCorresponding to the two values of the electron spin quantum number +1/2and -1/2The above quantity is called the Bohr magneton and is equal to: µB=µz=eh4!m= 9.27 "10#24A.m2The magnetic moment of the electron is the prime origin offerromagnetism in materials.22. The dipole moment associated with an atom of iron in an iron baris 2.1x10-23 J/T. Assume that all the atoms in the bar, which is 5.0 cmlong and has a cross-sectional area of 1.0 cm2, have their dipolemoments aligned.(a) What is the dipole moment of the bar? The number of iron atoms in the iron bar isN =7.9 g cm3( )5.0cm( )1.0cm2( )55.847 g mol( )6.022 ! 1023mol( )= 4.3 ! 1023.Thus, the dipole moment of the bar isµ= 2.1 ! 10"23J T( )4.3 ! 1023( )= 9.03 A # m2.(b) What torque must be exerted to hold this magnet perpendicular to an external field of 1.5 T? (The density of iron is 7.9 g/cm3)(b)!=µB sin 90o= 9.03A " m2( )1.5T( )= 13.5N " m(C) Use the dipole formula to find the magnitude and direction of themagnetic field 1cm from the end of the bar magnet on its central axis at P.µ= 8.9 !.m2 5 cmA = 1 cm2 B =µ0µ2!z3 B =4!"10#7NA28.9A.m2!" 0.05m(.01m)2 B =4 ! 10"78.9N / A.m5 ! 10"6 B = 0.71 TBT=µ0µ2!Ldzz3.06.01"= (µ0µ2!L)(#2z20.060.01) = (µ0µ2!L)(#2(0.01)2)  BT= dB!=µ02"dµz3!dµ=µALAdz =µLdzz.PBigBite is a 50 tonelectromagnet witha 25 cm by 100 cmgapB = 1 TeslaMaxwells Equations:In 1873 he wrote down 4 equations which govern allclassical electromagnetic phenomena.You already know two of them.  1. !E=! E .dA"= qenc/#0  2. !B=! B .dA"= 0A magnetic field changing with time can produce anelectric field: Faraday’s law  3. ! E .d! s != "d#BdtElectric lines curl around changing magnetic field linesLine integral of the electric fieldaround the wire equals the change of Magnetic flux through the areaBounded by the loopExampleNew Question: Can a changing electric field withtime produce an magnetic field?.Yes it can and it is calledMaxwell’s law of induction  ! B .d! s !=µ0"0d#EdtMaxwell’s law ofinduction  ! B .d! s !=µ0"0d#EdtConsider the charging of our circular plate capacitorB field also induced at point 2.When capacitor stops chargingB field disappears."# current ever actually flows through the capacitorFind the expression for the induced magnetic field Bthat circulates around the electric field lines of acharging circular parallel plate capacitor  ! B .d! s !=µ0"0d#Edt0BERr  ! B .d! s != (B)(2"r) since B parallel dsFlux withinthe loop of radius rr < R µ0!0d"Edt=µ0!0d(AE )dt=µ0!0AdEdt (B)(2!r) =µ0"0!r2dEdt B =µ0!0r2dEdtr< Rr > R B =µ0!0R22rdEdtAmpere-Maxwell’s Law  4. ! B .d! s !=µ0"0d#Edt+µ0ienc  ! B .d! s !=µ0"0d#Edt  ! B .d! s !=µ0iMaxwell combined the above two equations to form oneequationHow do we interpret this equation?This term has units of currentWhat is the displacement current?  ! B .d! s !=µ0"0d#Edt+µ0ienc !0d"Edt= idThis is called the displacement current id  ! B .d! s !=µ0id+µ0iencThe term is really is a transfer of electric and magnetic energy from one plate to the other while the plates are being charged or discharged. When charging stops, this term goes to zero. Note it is time dependent.Can I detect the magnetic field associated with displacement current? !0d"Edt= id  ! B .d! s !=µ0"0d#EdtShow that the displacement current in the gap of the twocapacitor plates is equal to the real current outside the gapCalculation of id E =!"0=q / A"0 q =!0AEFor the field inside a parallel plate capacitor i =dqdt=!0AdEdtThis is the real current i charging the capacitor.Solving for qFirst find the