Sources of the Magnetic FieldScheduleWatsgoinondisweek?Last time - Current LoopA length L of wire carries a current i. Show that if the wire is formed into a circular coil, then the maximum torque in a given magnetic field is developed when the coil has one turn only, and that maximum torque has the magnitude … well, let’s see.Problem continued…NEW TOPIC: SOURCES OF THE MAGNETIC FIELDRemember the wire?Try to remember…The “Coulomb’s Law” of MagnetismFor the Magnetic Field, current “elements” create the field.Magnetic Field of a Straight WireFrom the PastSlide 14Let’s Calculate the FIELDThe DetailsMoving right alongA bit more complicated A finite wireP1More P1P2Slide 22Center of a Circular Arc of a Wire carrying currentMore arc…The overall field from a circular current loopIronHowya Do Dat??Force Between Two Current Carrying Straight Parallel ConductorsThe CalculationDefinition of the AmpereAmpere’s LawRemember GAUSS’S LAW??Gauss’s LawThe next few slides have been lifted from Seb Oliver on the internetBiot-SavartInvisible SummaryMagnetic Field from a long wireSum B.ds around a circular pathConsider a different pathSO, AMPERE’S LAW by SUPERPOSITION:Slide 41The Right Hand Rule .. AGAINAnother Right Hand RuleCOMPARESimple ExampleField Around a Long Straight WireField INSIDE a Wire Carrying UNIFORM CurrentSlide 48Slide 49ProcedureThe figure below shows a cross section of an infinite conducting sheet carrying a current per unit x-length of l; the current emerges perpendicularly out of the page. (a) Use the Biot–Savart law and symmetry to show that for all points P above the sheet, and all points P´ below it, the magnetic field B is parallel to the sheet and directed as shown. (b) Use Ampere's law to find B at all points P and P´.FIRST PARTApply Ampere to CircuitThe “Math”A Physical SolenoidInside the SolenoidMore on Long SolenoidThe real thing…..Another WayApplicationTwo Coils“Real” Helmholtz CoilsThe ToroidSlide 64Sources of the Sources of the Magnetic FieldMagnetic FieldMarch 22, 2006March 22, 200610 13-Mar BREAK BREAK BREAK11 20-Mar Magnetic Field Sources of B Sources of B12 27-Mar Ampere’s Law Faraday's Law Faraday’s Law13 3-Apr Inductance InductanceEXAM #314 10-Apr Inductance - Finish AC Circuits AC Circuits15 17-Apr AC Circuits EM FIELDS Light Topics16 24-Apr Light Topics No ClassFINAL EXAM17 1-May No Class No Class No ClassScheduleWatsgoinondisweek?Watsgoinondisweek?Today we start looking at how Today we start looking at how currents create magnetic fields.currents create magnetic fields.Quiz on FridayQuiz on Friday–Magnetism ChapterMagnetism Chapter–NOT today’s materialNOT today’s materialNext Exam is on April 7Next Exam is on April 7thth..New WebAssign on Sources will be New WebAssign on Sources will be posted shortly.posted shortly.Last time - Current LoopBμAμandMoment Magnetic Define wire.of turnsN with coil aConsider sinNiiABA length L of wire carries a current i. Show that if the wire is formed into a circular coil, then the maximum torque in a given magnetic field is developed when the coil has one turn only, and that maximum torque has the magnitude … well, let’s see.Circumference = L/NNLrNLr22Problem continued…4and 1N when Maximum(BiA) ....... 42290 is angle the when maximum is B),sin( since 222222oiBLNLiBNNLiBNLNiBrANiABNEW TOPIC: SOURCES NEW TOPIC: SOURCES OF THE MAGNETIC FIELDOF THE MAGNETIC FIELDChapter 30Chapter 30Remember the wire?Try to remember…30204141rdqrdqrdrrEVECTOR UNITrrThe “Coulomb’s Law” of MagnetismA Vector Equation … duckFor the Magnetic Field,current “elements” create the field.TmATmtypermeabiliridridunit77030201026.1/10444:field fashion tosimilar aIn rsrsBEThis is the Law ofBiot-SavartMagnetic Field of a Straight WireWe intimated via magnets that the Magnetic field associated with a straight wire seemed to vary with 1/d.We can now PROVE this!From the PastUsing MagnetsRight-hand rule: Grasp the element in your right hand with your extended thumb pointing in the direction of the current. Your fingers will then naturally curl around in the direction of the magnetic field lines due to that element.Let’s Calculate the FIELDNote:For ALL current elementsin the wire:ds X ris into the pageThe Details02020)sin(2Bit. DOUBLE and to0 from integrate wesoamount equalan scontribute wire theofportion Negative)sin(4rdsiridsdBMoving right along RiRsRdsiBSoRsRRsr22)sin(sin002/322022221/dA bit more complicatedA finite wireP1)sin()sin(:NOTErds 2/12220)sin(4)sin()sin( RsrrdsidBrRrdsrdsMore P1 RiwhenRLLRiBandRsdsiBLL2B,L 42402202/2/2/3220P2 22002/322044RsLRiBorRsdsiRBLAPPLICATION: Find the magnetic field B at point PCenter of a Circular Arc of a Wire carrying currentMore arc…Cpoint at 444440000202020RiBdRiRiRddBBRiRdRidsdB Rdds dsThe overall field from a circular current loopSorta looks like a magnet!IronHowya Do Dat??0rsdNo Field at CForce Between Two Current Carrying Straight Parallel ConductorsWire “a” createsa field at wire “b”Current in wire “b” sees aforce because it is movingin the magnetic field of “a”.The CalculationdiLiiFdiBbaba2Fangles...right at are and Since2:calculatedjust what weis a"" wire todue b"" at wire FIELD The0b""on 0b""at BLBLDefinition of the AmpereThe force acting between currents in parallel wires is the basis for the definition of the ampere, which is one of the seven SI base units. The definition, adopted in 1946, is this: The ampere is that constant current which, if maintained in two straight, parallel conductors of infinite length, of negligible circular cross section, and placed 1 m apart in vacuum, would produce on each of these conductors a force of magnitude 2 x 10-7 newton per meter of length.Ampere’s LawThe return of GaussRemember GAUSS’S LAW??0enclosedqd -AESurfaceIntegralGauss’s LawMade calculations