Lecture 27Chapter 32Magnetism of MatterReview• 3 types of magnetism– Diamagnetism– Paramagnetism– Ferromagnetism• Explain how materials exhibit different types of magnetism using electron’s spin and orbital magnetic dipole moments• Placing material in an external B field causes dipole moments to align creating an induced Bfield • Degree of alignment determines type of magnetism and amount of magnetizationReview• Magnetic flux through an area A in a B field is • Induced emf occurs when magnetic flux changes with time • Changing B field induces an E field (Faraday’s law)dtdsdEBΦ−=•∫rr∫•=Φ AdBBrrdtdNBΦ−=EMagnetism (18)• Magnetic monopoles do not exist• Express mathematically as• Integral is taken over closed surface• Net magnetic flux through closed surface is zero– As many B field lines enter as leave the surface0=•=Φ∫AdBBrrMagnetism (19)• Gauss’s law for E fields• Gauss’s law for B fields• Both cases integrate over closed Gaussian surface0=•=Φ∫AdBBrr0εencEqAdE =•=Φ∫rr• Faraday’s law of inductionE field is induced along a closed loop by a changing magnetic flux encircled by that loop• Is the reverse true?• Maxwell’s law of inductionB field is induced along a closed loop by a changing electric flux in region encircled by loopMagnetism (20)dtdsdBEΦ=•∫00εµrrdtdsdEBΦ−=•∫rr• Consider circular parallel-plate capacitor with E field increasing at a steady rate• While E field changing, B fields are induced between plates, both inside and outside (point 1 and 2).• If E field stops changing, B field disappearsMagnetism (21)dtdsdBEΦ=•∫00εµrr• Two differences– Extra symbols, µ0and ε0, to preserve SI units– Minus sign – means induced E field and induced B field have opposite directions when produced in similar situationsMagnetism (22)dtdsdEBΦ−=•∫rrdtdsdBEΦ=•∫00εµrr•Ampere’s law• Combine Ampere’s and Maxwell’s law• B field can be produced by a current and/or a changing E field– Wire carrying constant current, dΦE /dt = 0– Charging a capacitor, no current so ienc = 0Magnetism (23)encEidtdsdB000µεµ+Φ=•∫rrencisdB0µ=•∫rrMagnetism (24)• What is the induced B field inside a circular capacitor which is being charged?• No current between capacitor plates so ienc = 0 and equation becomesencEidtdsdB000µεµ+Φ=•∫rrdtdsdBEΦ=•∫00εµrrMagnetism (25)• For left-hand side of equation chose Amperian loop inside capacitor• B and ds are parallel and B is constant so ∫∫=•θcosBdssdBrr)2(0cos rBdsBBdssdBπ===•∫∫∫rrMagnetism (26)• For right-hand side of equation find E flux through Amperian loop• E uniform between plates and ⊥⊥⊥⊥ to area A of loop • Right-hand side of equation becomesEAAdEE=•=Φ∫rrdtdEAEAdtddtdE000000)(εµεµεµ==ΦMagnetism (27)• Equating two sides gives• A is area of loop • Solving for B field inside capacitor givesdtdEArB00)2(εµπ=2rAπ=dtdErB200εµ=• B increases linearly with radius• B = 0 at center and max at plate edgesMagnetism (28)• What is the induced B field outside a circular capacitor which is being charged?• Realize ienc= 0 and find same relations encEidtdsdB000µεµ+Φ=•∫rr)2( rBsdBπ=•∫rrdtdEAdtdE0000εµεµ=ΦMagnetism (29)• E field only exists between plates so area of E field is not full area of loop, only area of plates • B field becomesdtdEArB00)2(εµπ=2RAπ=dtdErRB2200εµ=• Outside capacitor, B decreases with radial distance from a max value at r = RMagnetism (30)• Can represent change in electric flux with a fictitious current called the displacement current, id• Ampere-Maxwell’s law becomesdtdiEdΦ=0εencEidtdsdB000µεµ+Φ=•∫rrencencdiisdB0,0µµ+=•∫rrMagnetism (31)• Think of displacement current as fictional current between plates • Use right-hand rule to find direction of B field for both currentsencencdiisdB0,0µµ+=•∫rrMagnetism (32)• Used Ampere’s law to calculate B field inside a long straight wire with current i• Find B field inside a circular capacitor just replace i with displacement current, idencisdB0µ=•∫rrrRiB=202πµrRiBd=202πµMagnetism (33)• Used Ampere’s law to calculate B field outside a long straight wire with current i• Find B field outside a circular capacitor just replace i with displacement current, idencisdB0µ=•∫rrriBπµ20=riBdπµ20=Magnetism (34)• Checkpoint #6 – Parallel-plate capacitor of shape shown. Dashed lines are paths of integration. Rank the paths according to the magnitude of integral Bds when capacitor is discharging, greatest first.• Only displaced current in capacitor • What is idfor each path?encencdiisdB0,0µµ+=•∫rrencdisdB,0µ=•∫rrb, c, d all tie, then aMagnetism (35)• Basis of all electrical and magnetic phenomena can be described by 4 equations called Maxwell’s equations• As fundamental to electromagnetism as Newton’s law are to mechanics• Einstein showed that Maxwell’s equations work with special relativity• Maxwell’s equations basis for most equations studied since beginning of semester and will be basis for most of what we do the rest of the semesterMagnetism (36)Maxwell’s 4 equations are• Gauss’ Law• Gauss’ Law for magnetism• Faraday’s Law• Ampere-Maxwell Law0εencqAdE
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