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Chapter 29A whole picture helpsMathThe Magnetic FieldUnits of Magnetic FieldMagnetic field lines of bar magnets, shown by iron filings.What generates the magnetic field?Magnetic force onMagnetic force on moving chargeA few examplesCharged Particle in a Magnetic FieldForce on a Charged ParticleMore About Motion of Charged ParticleMotion of a Particle, GeneralVan Allen Radiation BeltsDifferences Between Electric and Magnetic FieldsMore Differences Between Electric and Magnetic FieldsNotation NotesCharged Particles Moving in Electric and Magnetic FieldsVelocity SelectorMass SpectrometerCyclotronCyclotronCyclotron, the calculationMagnetic Force on a Current Carrying Conductor, a wireForce on a Wire, the formulaTorque on a Current LoopTorque on a Current Loop, EquationTorque on a Current Loop, GeneralDirection of a current loop and the Magnetic Dipole MomentPotential EnergyHall Effect, a way to measure magnetic fieldHall voltage, negative (a) or positive (b) carriersHall voltage as a function of the magnetic fieldChapter 29Magnetic Fields1. Introduction to magnetic field.2. The forces on moving charges and currents inside a magnetic field.3. The math that will needed is vector product. 4. The source of magnetic field will be discussed in later chapters.A whole picture helpsCharge q as sourceCurrent I as sourceElectric field EMagnetic field BGauss’s LawAmpere’s LawForce on q in the fieldForce on or I in the filedrqvAmpere-Maxwell LawFaraday’s LawSummarized inMaxwell equations=r rqF E= �r rrqF v BMath= �r rrC A BVector cross product:θrArB= �r rrC A BVector cross product:= � =-r rr rD B A CθrArB= �rr rD A B( )sin� =rC ABθCDetermine the direction. IfMagnitude of the vector :rC= �r rrBF v BThe right-hand rule:1. Four fingers follow the first vector.2. Bend towards the second vector.3. Thumb points to the resultant vector.The Magnetic FieldThe field surrounds a magnet is called the magnetic field. The field is a vector, and is symbolized by Magnet exists in nature. Any magnets have two poles, called the north pole and the south pole. Like poles (from different magnets) repel, unlike poles attract.BrLike field lines in electric field, magnetic field lines are used to illustrate the field. Outside a magnet, field lines start from the north pole, end at the south pole. Field lines can be traced out by a small compass.Units of Magnetic FieldThe SI unit of magnetic field is the tesla (T)Wb is a weberA non-SI commonly used unit is a gauss (G)1 T = 104 G2( / )Wb N NTm C m s A m= = =� �Magnetic field lines of bar magnets, shown by iron filings.Field lines of one magnetof N and S polesof N and N polesMagnetic fieldElectric fieldComparison: there exist electric monopoles, the point charges. Magnet monopoles do not exist (have not been found). No matter how small a magnet is, it has two poles, N and S.What generates the magnetic field? Current (or moving charges, or changing electric field) generates magnetic field.We will get back to this topic in the following chapter.Magnet can take the form of a permanent magnet (ex. the bar magnet) or a solenoid. The Earth itself is also a big magnet.Magnetic force onMoving chargeCRTs (old TV tube)Particle acceleratorParticle mass spectrometerParticle detection and homeland securityCurrent carrying conductor.Electric motorHall effectA CRT (Cathode ray tube)A cyclotron acceleratorHere is the magnetic forceMagnetic force on moving chargeBq= �F v Br rrBFr is the magnetic fieldBrThe formula:q is the chargervis velocity of the chargeThe direction of the force is determined by the charge and the vector product of the velocity and the magnetic field. For a positive chargeThe magnitude:q=BF qvB sinor=BF qvBWhen the velocity and the field are perpendicular to each other.A few examplesCharged Particle in a Magnetic FieldConsider a particle moving in an external magnetic field with its velocity perpendicular to the fieldThe force is always directed toward the center of the circular pathThe magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particleUse the active figure to change the parameters of the particle and observe the motionPLAYACTIVE FIGUREForce on a Charged ParticleEquating the magnetic and centripetal forces:Solving for r:r is proportional to the linear momentum of the particle and inversely proportional to the magnetic field2BmvF qvBr= =mvrqB=More About Motion of Charged ParticleThe angular speed of the particle is The angular speed, , is also referred to as the cyclotron frequencyThe period of the motion is and this is not a function of the velocity.v qBωr m= =2 2 2πr π πmTvω qB= = =Motion of a Particle, GeneralIf a charged particle moves in a magnetic field at some arbitrary angle with respect to the field, its path is a helixSame equations apply, withUse the active figure to vary the initial velocity and observe the resulting motion 2 2y zv v v^= +PLAYACTIVE FIGUREVan Allen Radiation BeltsThe Van Allen radiation belts consist of charged particles surrounding the Earth in doughnut-shaped regionsThe particles are trapped by the Earth’s magnetic fieldThe particles spiral from pole to poleMay result in AurorasDifferences Between Electric and Magnetic FieldsDirection of forceThe electric force acts along the direction of the electric fieldThe magnetic force acts perpendicular to the magnetic fieldMotionThe electric force acts on a charged particle regardless of whether the particle is movingThe magnetic force acts on a charged particle only when the particle is in motionMore Differences Between Electric and Magnetic FieldsWorkThe electric force does work in displacing a charged particleThe magnetic force associated with a steady magnetic field does no work when a particle is displacedThis is because the force is perpendicular to the displacementProve:= � = � � =r rrr r0BdW d q dtF s v B vNotation NotesWhen vectors are perpendicular to the page, dots and crosses are usedThe dots represent the arrows coming out of the pageThe crosses represent the arrows going into the pageCharged Particles Moving in Electric and Magnetic FieldsIn many applications, charged particles will move in the presence of both magnetic and electric fieldsIn that case, the total force is the sum of the forces due to the individual


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SMU PHYS 1304 - Magnetic Fields

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