26. Magnetism: Force & FieldWhat is Magnetism?Magnetic Force & FieldCharged Particles in Magnetic FieldsThe Magnetic Force on a CurrentOrigin of the Magnetic FieldMagnetic DipolesMagnetic MatterAmperes law
This ultraviolet image shows delicate loops of million-kelvin ionized gas in the Suns atmosphere.
What force shapes the gas into such intricate structures, and why dont we see similar things in Earths atmosphere?
26.1. What is Magnetism?Magnets exert (magnetic) forces on each other & materials like irons.Iron filings align with the magnetic field, tracing out the field of a bar magnet.Field description:A magnet produces a magnetic field B.Another interacts with B at its position.Ampere :Moving charges can produce B.Magnetic dipoles ~ current loops.Quantum mechanics:Intrinsic magnetic moment related to spin.
26.2. Magnetic Force & FieldMagnetic force on a charge q moving with velocity v in a field B:[B] = N s / (C m)= T (Tesla)= 10,000 G (Gauss)BEarth ~ 1 GBsmall magnet ~ 100 GBMRI ~ 1 TBmagnetar ~ 1011 Tv B: greatest Fv // B:
GOT IT? 26.1Figure shows a proton in a magnetic field.For which of the three proton velocities shown will the magnetic force be greatest?What will be the direction of the force in all three cases? & into paper
Example 26.1. Steering ProtonsFigure shows 3 protons entering a 0.10-T magnetic field.All three are moving at 2.0 Mm/s.Find the magnetic force on each.velocity selector. Electromagnetic forceB123FB + FE = 0 when v = E/B
26.3. Charged Particles in Magnetic Fields WB = 0For v B, FB = q v B. Circular motionv = const CW about B for q > 0.
GOT IT? 26.2A uniform magnetic field points out of this page.Will an electron thats moving in the plane of the page circle clockwise or counterclockwiseas viewed from above the page?
Example 26.2. Mass SpectrometerA mass spectrometer separates ions according to their ratio of charge to mass.Such devices are widely used to analyze unknown mixtures,and to separate isotopes of chemical elements.Figure below shows ions of charge q & mass m being first accelerated from rest through a potential difference V,then entering a region of uniform B pointing out of the page.Find an expression for the horizontal distance x.
The Cyclotron FrequencyPeriod of particle in circular orbit in uniform B:Cyclotron frequencyTrajectory in 3-DCharged particles frozen to B field lines. Motion // B not affected by it
Application: The CyclotronWhole device in vacuum chamber.Small V across the dees, which alternates polarities at the cyclotron frequency.Particle injected at center of gap & spirals out.E ~ MeVs.Applications:Manufacture of radioactive isotopes.e.g., PET (Positron Emission Tomography).Higher energies: Relativity effects Synchrotron (B also varies)
26.4. The Magnetic Force on a CurrentForce on carrier in wire:Force on straight wire of cross section A & length L:Prob 58F out of papere moving left deflected upward resulting charge separation leads to upward force on whole wireFmag on + charge is also upward
GOT IT? 26.3Figure shows a flexible wire passing through a magnetic field that points out of page.The wire is deflected upward, as shown.Is the current flowing to the left or to the right?
Example 26.3. Power LineA power line runs along Earths equator, where B points horizontally from north to south and has strength 50 T.The current in the line is 500 A, flowing from west to east.Find magnitude & direction of the magnetic force on each kilometer of the power line.upwardc.f. weight of wire is ~ 10 kN
The Hall EffectDirection of FB depends on I, not on sign of charged carriers.Carriers of both signs are deflected upwardsDirection of E due to built-up charges depends on signs of charged carriers:Hall effect.Steady state, Fz = 0 :Hall potential:Hall coefficient:e moves to left & deflected upwardp moves to right & deflected upwardEH is upwardEH is downwardxyz
26.5. Origin of the Magnetic FieldBiot-Savart Law:permeability constantThumb // ICurl of finger gives B.C.f.
Example 26.4. Current LoopFind the magnetic field at an arbitrary point P on the axis of a circular loop of radius a carrying current I.By symmetry, only Bx 0.dL rcos = a / r
Example 26.5. Straight LineFind the magnetic field produced by an infinitely long straight wire carrying current I.
The Magnetic Force Between ConductorsField of I1 at I2 :Force on length L of I2 :Force per unit length on I2 :points toward I1Hum from electric equipments are vibrations of transformers in response to AC.Definition: 1 A is the current in two long, parallel wire 1 m apart & exerting 2107 N per meter of length.1C is the charge passing in 1s through a wire carrying 1A.
GOT IT? 26.4A flexible wire is wound into a flat spiral as shown in figure.If a current flows in the direction shown, will the coil tighten or become looser?Does your answer depend on the current direction?No
26.6. Magnetic DipolesField on axis of current loop of radius a :C.f. electric dipole:Setting magnetic dipole N-turn current loop:Detailed calculations show: = I A valid for arbitrary loop. Vector behavior of similar to that of p for r >> a.Far away, fields look similar but close in, theyre different.
Sunspots are regions of intense magnetic field. The large sunspot group below the solar equator was responsible for some of the most energetic solar outbursts ever recorded, which occurred in late 2003.Multi-turn loops = electromagnetsVery strong field requires superconducting wires, e.g., MRI scanners.Orbiting e in atom .Currents flows in Earth liquid outer core geomagnetic field,which deflects harmful high E solar particles.Magnetic reversal:Earth (period ~ millions of yrs) : map for sea-flooe spreading over.Sun (period ~ 11 yrs.) : sunspots.
Dipoles & MonopolesSome elementary particle theories suggest existence of magnetic monopoles.But none was ever observed. Microscopic origin of B:Charged current.Intrinsic spin.No magnetic charges : B lines always form loops,either encircling moving charges, or joining the 2 poles of a magnet.Gauss Law
GOT IT? 26.5The figure shows two fields. Which could be a magnetic field?
Torque on a Magnetic DipoleTorque on dipole:Associated energy:Forces on top & bottom cancel.Forces on sides also cancel; but give net torque.beneath planeabove plane
Application: Electric MotorsRotating loopBrushesCommutatorBatteryCCWCWAlways CCW
Example 26.6. Hybrid Car MotorToyotas Prius gas-electric hybrid car uses a 50-kW electric motor that develops a maximum torque of 400 Nm.Suppose you want to produce this same torque in the motor just discussed,with a 950-turn rectangular coil of wire measuring 30 cm by 20 cm,in a uniform field of 50 mT.How much current does the motor need?Max torque at sin = 1 :Rectangular coil :950-turn coil
26.7. Magnetic MatterFerromagnetism:e.g., Fe, Ni, Co, & their alloys.Material divided into magnetic domains in which s are aligned.T > TC : orientations of in different domains random < > B (paramagnetic)T < TC : orientations of in different domains aligned < > 0 even when B = 0 (ferromagnetic)Atomic current loops are CCW.Adjacent loops cancelNo cancellation on boundaries:net I on surface.
// Bappliedp // EappliedDistant field similar.Internal field opposite.Field inside dipole < EappliedField inside dipole > Bapplied
Paramagnetism :Materials with randomly oriented permanent and very weak - interaction. < > = B with > 0 = magnetic susceptibilityDiamagnetism :Materials with no intrinsic . < > = B with < 0( induced )
26.8. Amperes lawField around long wire carrying current I :from Biot-Savart lawTrue for arbitrary closed paths & steady currents:Amperes lawc.f. Gauss lawnet field from ALL sourcesBdr = 0
Example 26.7. Solar CurrentsThe long dimension of the rectangular loop in figure is 400 Mm,and B near the loop has a constant magnitude of 2 mT.Estimate the total current enclosed by the rectangle.B dr on the short segment B dr = 0 there.B // dr on the long segment Direction: into the page3D: around equator
GOT IT? 26.6The figure shows three parallel wires carrying current of the same magnitude I, but in one of them the current is opposite to that of the other two.If around loop 2,(a) what is around loop 1, and(b) which current is the opposite one? A0
Using Amperes LawBase on symmetry, choose the amperian loop such that B is either // or to it. STRATEGY 26.1Ampres Law :
Example 26.8. Outside & Inside a WireA long, straight wire of radius R carries a current I distributed uniformly over its cross section.Find the magnetic field outside and inside the wire.Amperian loop is a circle. (a)(b)By symmetry, B is azimuthal.
Example 26.9. Current SheetAn infinite flat sheet carries current out of this page.The current is distributed uniformly along the sheet, with current per unit width given by JS .Find the magnetic field of this sheet.Amperian loop is a rectangle. By symmetry, B is // to sheet & I. Far out, B lines nearly circularClose in, B lines ~ infinite sheet
Fields of Simple Current DistributionsFor arbitrary distributions:Far away from any loop ~ dipole.Very near any wire ~ infinite straight wire.Very near any current sheet ~ infinite flat sheet.
SolenoidsSolenoid: a long, tightly wounded coi