Magnetic Field And Force - Northeastern University

Magnetic field and forceSo far we have studied two forces: gravity and electricityMagnetism is a new force, but also related to electric charges.Gravity is created by mass and gravity acts on masses.vElectric fields are created by electric charges E And Electric fields exert forces on charges.vkQrˆr2vFE qEMagnetic field and forceThere is a different kind of field, called a magnetic field, orB-fieldvBB-fields are created by moving charges (currents).B-fields exert a force on moving charges.This is very different from our previously studied forces!1

Natural magnetismNatural magnetism2

What is the origin of this force?Magnetic charge? Monopoles?Scientists have postulatedthat there might be someother kind of object thathas a “magnetic charge” Magnetic Monopole.No magnetic monopoleshave ever been found.Thus, we do not considerthem in classes.3

Magnetism and electric chargeOersted’s experiment:A compass is placeddirectly over a wire (hereviewed from above). Apassing current deflectsthe needle.Force on a moving chargeWe will see later exactly how B-fields are made by movingcharges.Now we will study how B-fields exert forces on movingcharges.vv vF qv BThe magnetic force exerted on a charge q moving withvelocity v in a magnetic field B.This equation actually defines the magnetic B-field.4

Visualizing a M. Field: Field lineshttp://www.youtube.com/watch?v wuA-dkKvrd0Force on a moving chargevv vF qv BVector cross product.vv v F q v B sin θWhat about the resulting force direction fromthe cross product?Bθv5

Vector cross product – Right Hand Rulev vu v Fingers in direction of first vector. Bend them into direction of secondvector. Thumb points in cross product direction. If your hand does not bend that way, flipit around! – never use the left hand ;-)New vector direction is alwaysperpendicular to the original vectors!Force on a moving charge6

Consequences:vv vF qv Bvv v F q v B sin θ If particle is not moving (v 0) then no force. If velocity and B-field are parallel, then no force. If velocity and B-field are perpendicular, then maximum force. If q is negative, then force is in opposite direction.Drawing conventionsDrawing vector directions on a 2 dimensional piece of paper?Magnetic Field Vector to the right.Magnetic Field Vector out of the page.Arrow head pointing at you.XMagnetic Field Vector into the page.Arrow tail pointing at you.7

Clicker QuestionA negative particle and a positive particle are moving withcertain velocities in a constant, uniform magnetic field, asshown. The direction of the B-field is to the right. The ( )particle is moving directly left; the (–) particle is movingdirectly up.The force on the positive particle dueto the B-field is (in into page, out out of page).A: inB: outC: zeroD: rightE: leftBAnswer: The ( ) particle is moving anti-parallel to the B-field.The angle θ is 180 and the force is FB qvB sinθ 0.Clicker QuestionA negative particle and a positive particle are moving withcertain velocities in a constant, uniform magnetic field, asshown. The direction of the B-field is to the right. The ( )particle is moving directly left; the (–) particle is movingdirectly up.The force on the negative particledue to the B-field isA: inB: outC: zeroD: rightE: leftBAnswer: The (–) particle is moving at right angles tothe field. By the right-hand rule, the direction "vcross B" is into the page, but the particle has anegative charge q, so the force is out of the page.8

Clicker QuestionA positive particle is released from rest in a region of spacewhere there is constant, uniform, electric field and a constant,uniform magnetic field. TheCBAelectric field points up and the E.B(out)magneticfield points out of thepage in the diagram below.Which path will the positiveparticle follow? (All pathsshown are in plane of the page.)D: it will not moveAnswer: The ( ) particle will feel a force FE qE due to the Efield along the direction of the E-field. As it starts movingalong the E-field direction, it will acquire a velocity, and it willstart to feel a force FB qvB, due to the B-field. The direction ofthe force is to the right, by the right-hand-rule.Clicker QuestionA negative particle and a positive particle are moving withcertain velocities in a constant, uniform magnetic field, asshown. The direction of the B-field is to the right. The ( )particle is moving directly left; the (–) particle is movingdirectly up.The force on the negative particledue to the B-field isA: inB: outC: zeroD: rightE: leftBAnswer: The (–) particle is moving at right angles tothe field. By the right-hand rule, the direction "vcross B" is into the page, but the particle has anegative charge q, so the force is out of the page.9

Magnetic field unitsUnits for Magnetic Fieldvv vF qv BB-field [B] [Newtons] / [Coulomb xmeters/second] [Tesla]How big is a 1 Tesla Magnetic Field?Interstellar SpaceHuman BeingEarth’s SurfaceSun’s SurfaceSmall Bar MagnetExperiment MagnetMaximum Steady MagnetMaximum in Explosive MagnetSurface of Neutron Star10-10 Tesla10-10 Tesla5x10-5 Tesla10-2 Tesla10-2 Tesla1 Tesla30 Tesla1000 Tesla108 Tesla10

Another Unit System1 Gauss 10-4 TeslaThus, the Earth’s magnetic field is 0.5 Gauss.This unit system is often used when talking about smallmagnetic fields, but it is not the SI unit system!Earth’s magnetic field11

Clicker QuestionHere is an event display from ahigh energy experiment. There is a1 Tesla uniform magnetic fieldcoming out of the page. What isthe sign of the electric charge?A)PositiveB)Negativevv vF qv BVideosFeymann's "why" and "Fields"12

Motion of charged particles in a M. FieldBecause the force is alwaysperpendicular to the velocity (directionof motion), the Magnetic force can dono work on q.v vW F r 0sincevv vF qv BvvFB rB-field cannot change the KineticEnergy of a moving particle, butcan change its direction ofmotion.Wnet KE 0Charged particle in a perpendicular fieldvResults in circular motion!BvBParticle moving in a plane with a Bfield uniformly out of the plane.FNo change in KE, but constantchange in velocity direction.F13

Charged particle in a perpendicular fieldvvvvvv F m a qv B v F q v B Since velocity and B are alwaysperpendicular.mv 2Since circular motion. F q v B Rmv Radius of circular motionR depends on m, v, q, B.qBvvvvRBCharged particle in a perpendicular fieldOne can then solve for the frequency ofrevolution (“cyclotron frequency”).Rvf # revolutions/secondBqRB distance 2πR v PeriodmtimeT1qBThe cyclotron frequency isf independent of R.T 2πm14

Helical motionWhat if in addition to thecircular motion in this plane,there is a non-zero velocityout of the page?vRBR mvqBThis extra velocitycontributes no additionalforce since it is parallelto B.Aurora borealis (“Northern lights”)15

Clicker QuestionA ( ) charged particle with an initial speed vo is moving in aplane perpendicular to a uniform magnetic field (B into thepage). There is a tenuous gas throughout the region whichcauses viscous drag and slows the particle over time. Thepath of the particle isBA: a spiral inwardB: a spiral outwardC: something elseApplications: Mass spectrometer16

Applications: velocity selectorFE qEElectric force:Magnetic force:Total force:For the particle to pass and not be deflected:FB qvB F qvB qEqvB qE 0 v EBThomson’s e/m experimentFrom energy conservation:We have seen that to passthrough we need:Combining the two we obtain:1 22eVmv eV v 2mEv BeE2 m 2VB 217

Force on current carrying wiresSince B-fields exert forces on movingcharges, it is natural that B-fields exertforces on current carrying wires.vv vF qv BHow do we quantify this force in termsof current I, instead of q and v?Force on current carrying wiresConsider the wire shown below.Now add a Magnetic Field.ForceXArea A vv# charges in thissegment of wireBN Velocity vnxAxL#/VolumeLength vector LvF (one charge) qv BvVolumevvF (total ) (nALq )v B18

Force on current carrying wiresvvvF (total ) ( nALq)v BHow to relate this to current?J I nqvdAvvvF (total ) IL BI nAqvd* Notice that L points along v, whichis the direction of the current.Clicker QuestionA current-carrying wire is in a B-field. The wire is oriented tothe B-field as shown. What is the direction of the magneticforce on the wire?A) RightB) DownC) Out of the PageD) Into the PageE) None of these.Bi19

Force on a straight wirevvvF (total ) iL BForceiXBExampleForce on electric wires due to Earth’s Magnetic FieldPower line of 1000 meters runs along theEarth’s equator where the B-field 0.5Gauss points South to North.The current in the wire is 500 Amps goingEast to West.vvvF (total ) IL BvF (500 A)(1000m)(0.5 10 4 T ) 25 N upWeight of the wire 20,000 Newtons.20

General expression for the force on a wireIf the wire is not straight or the B-field is not uniform, we needto break the wire up into little segmentsvidFvdLX BvvdF idL BvvvFtot dF idL BClicker QuestionA square loop of wire carrying current I is in a uniformmagnetic field B. The loop is perpendicular to B (B out of thepage). What is the direction of the net force on the wire?A: out of the pageB: into the pageC: D: E: None of theseBI21

The DC motorWe can see (Section 27.7) that a magnetic field canproduce torque on a look of wire carrying a currentMagnetic Field Lines and Flux22

Gauss’s Law for Magnetic FieldsMagnetic monopoles (so far) do not exist!!!There are no sources of magnetic flux Flux through aclosed surface:rr B dA 023

Small Bar Magnet 10-2 Tesla Experiment Magnet 1 Tesla Maximum Steady Magnet 30 Tesla Maximum in Explosive Magnet 1000 Tesla Surface of Neutron Star 108 Tesla. 11 Another Unit System 1 Gauss 10-4 Tesla Thus, the Ear