Chapter 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 filedqvAmpere-Maxwell LawFaraday’s LawSummarized inMaxwell equations= qF E= × qF v BMath= × C A BVector cross product:θAB= × C A BVector cross product:= × = − D B A CθAB= × D A B( )sin≡ =C ABθCDetermine the direction. IfMagnitude of the vector :C= × BF 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 FieldThe 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.BLike 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 FieldThe SI unit of magnetic field is the tesla (T)Wb is a weberA non-SI commonly used unit is a gauss (G)1 T = 104G2( / )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 onMoving chargeCRTs (old TV tube)Particle acceleratorParticle mass spectrometerParticle detection and homeland securityCurrent carrying conductor.Electric motorHall effectA CRT (Cathode ray tube)A cyclotron acceleratorHere is the magnetic forceMagnetic force on moving chargeBq= ×F v B BFis the magnetic fieldBThe formula:q is the chargevis 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:θ=BF qvB sinor=BF qvBWhen the velocity and the field are perpendicular to each other.A few examplesCharged Particle in a Magnetic FieldConsider a particle moving in an external magnetic field with its velocity perpendicular to the fieldThe force is always directed toward the center of the circular pathThe magnetic force causes a centripetal acceleration, changing the direction of the velocity of the particleUse the active figure to change the parameters of the particle and observe the motionPLAYACTIVE FIGUREForce on a Charged ParticleEquating 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 ParticleThe angular speed of the particle is The angular speed, ω, is also referred to as the cyclotron frequencyThe 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, GeneralIf a charged particle moves in a magnetic field at some arbitrary angle with respect to the field, its path is a helixSame equations apply, withUse the active figure to vary the initial velocity and observe the resulting motion 2 2y zv v v⊥= +PLAYACTIVE FIGUREVan Allen Radiation BeltsThe Van Allen radiation belts consist of charged particles surrounding the Earth in doughnut-shaped regionsThe particles are trapped by the Earth’s magnetic fieldThe particles spiral from pole to poleMay result in AurorasDifferences Between Electric and Magnetic FieldsDirection of forceThe electric force acts along the direction of the electric fieldThe magnetic force acts perpendicular to the magnetic fieldMotionThe electric force acts on a charged particle regardless of whether the particle is movingThe magnetic force acts on a charged particle only when the particle is in motionMore Differences Between Electric and Magnetic FieldsWorkThe electric force does work in displacing a charged particleThe magnetic force associated with a steady magnetic field does no work when a particle is displacedThis is because the force is perpendicular to the displacementProve:= ⋅ = × ⋅ = 0BdW d q dtF s v B vNotation NotesWhen vectors are perpendicular to the page, dots and crosses are usedThe dots represent the arrows coming out of the pageThe crosses represent the arrows going into the pageCharged Particles Moving in Electric and Magnetic FieldsIn many applications, charged particles will move in the presence of both magnetic and electric fieldsIn that case, the total force is the sum of the forces due to the individual fieldsIn general (The Lorentz force): q q= + ×F E v B Velocity SelectorA uniform electric field is perpendicular to a uniform magnetic fieldWhen the force due to the electric field is equal but opposite to the force due to the magnetic field, the particle moves in a straight lineThis selects particles with velocities of the value v = E / BPLAYACTIVE FIGURESlitMass SpectrometerA mass spectrometer separates ions according to their mass-to-charge ratioA beam of ions passes through a velocity selector and enters a second magnetic field where the ions move in a semicircle of radius r before striking a detector at P. From and v from the velocity selector, the mass m of the particle is measured.If the ions are positively charged, they deflect to the left; If the ions are negatively charged, they deflect to the rightPLAYACTIVE FIGUREmvrqB=CyclotronA cyclotron is a device that can accelerate charged
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