Survey Of Magnetism And Electrodynamics (Chapters 5-11)

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Survey of Magnetism and Electrodynamics (Chapters 5-11)* Electrostatics - Coulomb’s lawclosedpotential ”flow “surfaces* Magnetostatics - Biot-Savart lawV 1I 1V 2I 2discontinuity I 0I 6 !I 6closed fluxlines Bdivergent field”flux “lines Dfrom source QI 5I 3I 4curling flowsurfaces of Haround source I* Helmholtz theorem: source and potential* Green ‘s function (tent/pole)* Macroscopic media - polarization chains* magnetization chainsfree charge in conductorbound chargein dielectricM-chain(solenoid)magneticcoil* Faraday’s law* Maxwell’s displacement current

* Three electrical devices, each the ratio of flux / flowCAPACITORRESISTOR* Electrodynamics equationsINDUCTORLorentz forceContinuityMaxwell electric,magnetic fieldsConstitutionPotentials(wave equation)* Conserved currentsdesity fluxenergymomentum* Electromagnetic waves- Fresnell’s coefficients- skin depth- dipole radiationGauge transform

Section 5.1.1 – Magnetic Fields* the magnetic force was known in antiquity, but was more difficult to quantify predominant effect in nature involves magnetization, not electric currents no magnetic (point) charge (monopole); 1-d currents instead of 0-d charges static electricity was produced in the lab long before steady currents* History: http://maxwell.byu.edu/ spencerr/phys442/node4.html600 BC1200 AD1259 AD1600 AD1742 AD1820 AD1820 AD1820 ADThales of Miletus discovers lodestone’s atraction to ironChinese use lodestone compass for navigationPetrus Peregrinus (Italy) discovers the same thingWilliam Gilbert discovers that the Earth is a giant magnetThomas LeSeur shows inverse cube law for magnetsHans Christian Oersted discovers that current twists magnetsAndre Marie Ampere shows that parallel currents attract/repelJean-Baptiste Biot & Felix Savart show* for magnetism it is must more natural to start with the concept of field compass points Nbecause it aligns withthe Earth’s magnetic field iron filings chain up toshow physical “field lines” bar magnet fieldlines resemble anelectric dipole* what is the main difference? two differences related to “flux” and “flow” difference between “internal” and “external” dipoleint.Amber (electric) rub to chargedirect force2 charges /fluid monopolesLodestone (magnet) always chargedtorque2 poles (N/S)unseparable dipoleint.ext. sources/sinks of flux conservative flow (potential)ext. lines of flux conserved flow is rotational* no magnetic monopole! this is the source of thedifferences between E vs. B eq.and dielecrics vs. magnets N/S poles cannot be separated reason: magnetic dipoles are actually current loops note: field lines are perpendicuar to source current* discovery by Hans Christian Oersted (1820) current produces a magnetic field generalized to the force between wires by Ampere, Biot and Savart* Ampere for two wires separated by distance d definition of Ampere [A] definition of Tesla [T]and Gauss (CGS units)Coulomb [c]

Section 5.1.2 – Lorentz Force, Current Elements* magnetic force law the combinationoccurs frequently, it is called the “current element” units: A m C m/s qv, much like a “charge element”* current density* conservation of charge: Kirchov’s current law written as 4-vectors* relation between charge and current elements* Lorentz force lawis any ”charge element “is any ”current element “* Magnetic forces do no worktangential acceleration (not quite)radial acceleration(always)”gas pedal “”steering wheel “ similar to the normal force which only deflects objects

Section 5.2 – Biot-Savart Law* review: charge element (scalar): current element (vector!):surface, volume current density steady currents: analog ofelectrostatic stationary charges* electrostatic vs. magnetostatic force laws definition of “force” fields E, B(vs. “source” fields D, H, see next chapter) fields mediate force from one charge (current) to another (action at a distance) experiment by Oersted defined direction of field, Ampere defined magnitude Coulomb Law (electric) Biot-Savart Law (magnetic) proof: combined: Lorentz force law* Example 5.5: Parallel wiresdirectionalinverse-squareforce law for an infinite wire: for a second parallel wire:as shown before, this was unsed to define the Ampere (current) - Tesla (B-field)

* Example 5.6: Current loop:* Example: Off-axis field of current loop:

Section 5.3 – Div and Curl of B* the formalism of both electrostatics and magnetostatics follow the Helmholtz theorem* these two diagrams illustrate the symmetry between the two forcesForceForceCoulomb & SuperpositionBiot-Savart & SuperpositionHelmholtzHelmholtzIntegral eld eq.Di erential eld eq.GaussIntegral eld eq.Di erential eld ialLaplaceLaplaceGreenGreen* integral equations assumes exactly 1 winding, otherwise, by superposition,* differential equationsMomentum

Applications of Ampere’s law* Ampere’s law is the analog of Gauss’ law for magnetic fields uses a path integral around closed loop instead of integral over a closed surface simplest way to solve magnetic fields with high symmetry* Example 5.7: straight wire* Example 5.8: current septum* Example 5.9: infinite solenoid winding density #turns / length* Maxwell’s equations (steady-state E&M) the two zeros mean there is no magnetic monopole actually as long asis constant, a magnetic monopole canturned into an electric charge by a redefinition of E amd B (duality rotation)

Section 5.x – Magnetic Scalar Potential* pictorial representation of Maxwell’s steady state equations defineto drop all the‘s and emphasize the “source” aspect of flux: flow:electricmagneticelectricmagnetic* utility of treating D,B as flux linesand E,H as equipotential surfaces: flux through a surface S # of lines that poke through a surface S flow along a curve/path P # of surfaces that a path P pokes through* potentials, from Helmholtz theorem* scalar electric and magnetic potentialsALWAYSONLY ifPoisson’s eq.Laplace’s eq. ALWAYS solvewith appropriate B.C.’s* boundary conditions electric magnetic surface current flows along U equipotential U is a SOURCE potential the current I I2-I1 flows between any two equipotential lines U I1 and U I2

Examples* capacitor* solenoid* distcontinuities in U* toroida) at I: the edge of each H sheet is an I lineb) around I: the U 0,6 surfaces coincidea “branch cut” on U extends from each I line U is well defined in a simply connected regionor one that does not link any current* procedure for designing a coil based onrequired fields and geometry: solvein the with flux boundaryconditions from known external fields the equipotential CURVES along the boundary surface become windings the current through each winding is the difference between each equipotential* utility of electric and magnetic potentials it is only possible to control electric potential, NOT charge distribution in a conductor is also practical to control the magnetic potential ( current distribution) using wires* cos-theta coil cylindrical (2-d) analog of electric dipole longitudinal windings, perfectly uniform field inside solve Laplace equation with flux boundary condition double cos-theta coil B 0 outside outer coil

Section 5.4 – Magnetic Vector Potential* Helmholtz theorem* Gauge invariance: A is NOT unique! Onlysospecified, not(Helmholtz)also satisfies is called a “gauge transformation”, the set of all‘s forms a mathematical groupsymmetry under gauge transformations is the basis of quantum field theories a particular choice of A or a constraint on A is called a “gauge” “Coulomb” or “radiation” gauge:always possible, unique up to B.C.’sifletand solve for(another Poission eq.)* Boundary conditionsH links current,A links flux* Physical significance: qV potential energy qA “potential momentum” it is the energy/momentum of interaction of a particle in the field some special cases can be solved using conservation of momentum,but you must account for momentum of the field unless there are no gradients (V,A) is a 4-vector, like (E, p ) (c,v) ( ,J) q(V- v A) is a velocity-dependent potentialAmerican Journal of Physics 64, 1368 (1996)

Examples* B flux tube (solenoid)W i reS ole noid(inside)(outside)* Coaxial cable, straight conductorAmerican Journal of Physics 64, 1368 (1996)

Section 5.4.3 – Multipole Expansion* Similar to electrostatics, expand 1/rno monopoledipolequadrupoledivergence theoremsletthenletthencompare:* in spherical coordinates,compare:

* Example: current loop dipole the above integral is antisymmetric under to get the dipole approximation, assume the first term first term vanishes- no monopole! the second two terms are the dipole equivalent to electric dipole under correspondence* Summary of vector axwell eq.’sPoisson’s eq.conservationof charge

Section 6.1,2 – Magnetization* dynamics of dipoles in fields (compare Electric and Magnetic) electric force magnetic force electric torque magnetic torque* polarizabilityelectrica) stretch /- chargeb) torque on permanent dipolesmagnetica) torque on spinb) speed up orbitalc) self-alignment of dipolesparamagneticdiamagneticferromagnetic* magnetization* field of a magnetized object: bound currents generalized divergence theoremor scalar mult.where notice the difference in signscompare

Section 6.3 – Auxiliary Field H* free current in general,can be vector functions of H field (vector and position, temp, history) compare with electric:fluxpotentialpolsourceforcepoleven with“magnetic charge!” in Gilbert picture flux tube with a source and sink at each end,or a N-S chainpresent

* Example - capacitorvssolenoid or permanent magnetfor long skinny magnet,* Example - spherical permanent magnetic

Section 6.4 - Magnetic Media* constitutive relations: magnetic susceptibility and permeability* linear and nonlinear media: linearpermeability independent of field strength isotropicsame permeability in all directions homogeneoussame permeability throughout materialstill dicontinuities at boundary* Gaussian units (CGS) [H] Oersted, [B] Gauss 0.0001 Tesla Units of E and B also the same!* diamagnetism most similar to electiric useful for levitation superconductor (SC)* paramagnetism* ferromagnetism electromagnet iron-coretransformers -metal1.8remanenceB (T)BR1.21.7 T1.5 T1.2 T1.0 T0.8 T0.5 T0.3 T0.60coercivityHC-0.6-1.2-1.8 150H (A/m) 100 50050http://en.wikipedia.org/wiki/Hysteresis100150

* Example - magnetic sphere in external field

Section 7.1 - Electromotive Force* review current element continuity potentialconservation of chargeconservation of energy* conductors static case steady current RESISTOR what if current(third constitutive equation) resistor vs. CRTresistivityconductivity Drude law: bumper carstime between collisionsmean free pathatomic density x # carriers/atom* power dissipation* relaxation time versus CAPACITOR

* electromotive force (emf) electromotance more correct! compare: magnetomotance (HW4, #3) forces on electrons from E and other sources (chemical, B, .) not quitesince 0generalization of(emf )* motional emf - magnetic forces relation to flux: precursor to Faraday’s law conservation of energy: magnetic force does no work! general proof

Section 7.2.1 - Faraday’s Law* three experiments – one result!a) moving loop in static B field (7.1)b) static loop in moving B fieldc) static loop in static changing B fieldchange of(nonuniform field)referencemotion of fluxframe (S.R.)lines irrelevantonly net flux motional emf Faraday’s law* different physics involved, both involving B fieldsa) Lorentz force lawb,c) Faraday’s law- static charge in changing field- moving charge in static field* Special Relativity* Lenz’s law equivalence of E&M in different ref. frames fields have “inertia” Lorentz transformations, it takes energy to build/destroy E,Bboth components of currents oppose change in fields* Example of a) - AC generator 3-phase generator has 6 maxima of current per cycle both 1-phase and 2-phase only have 2 bicycle pedal problemI* Example 7.5t* Example 7.6

Section 7.2.2 - Induced Electric Field* three Ampere-like laws - one technique!AmpereVector Potential* with proper symmetry, each can be solved with Amperian loop* Example 7.8: charge glued on a wheel angular momentum from turning off field independent of time alternate approach: vector potential (momentum)* Problem 7.12: mutual inductance‘reluctance’ ‘permeance’Faraday

Section 7.2.3 - Inductance* review: 3 ”Ampere “ laws will use all 3 today* new V I ”Z “ lawcurrenttimefluxvoltage* Inductance - application of Faraday’s law property of material and geometry ”back “ emf: voltage drop across L,opposes changes in the current* Inductance matrix L symmetric: mutual inductance diagonal:self inductance* three electrical devices - one calculation!* unitscompare:

Section 7.2.4 - Energy in the Magnetic Field* example: L-R circuit time constantnote: initial slope depends on L, not Rlarger R just means lower I* work against back emf: ”electrical inertia “compare:energy from”potential momentum “* example 7.13

Section 7.3, 10.1 - Maxwell’s Equations* towards a consistent system of field ity* 2 problemsa) potentialsgauge invarianceb) continuity” displacement current “* example: capacitor - continuity: Ampere’s law should not dependon surface to integrate charge flux field should also exist in capacitor each new charge on platebuilds up a new D-flux line charge ”propagates“ through capacitorvia its associate D-flux line ”displacement current “:I flowing through wire D building up in capacitor* expand D, H” displacement current “

* Maxwell’s Eq’s in vacuum* integral form* boundary conditions - integrate Maxwell’s equations over the surfaceFieldsIngegralPotentials* duality transformation - another symmetry of Maxwell’s equations without sources, B E symmetry, except units symmetry with sources by adding magnetic charge (monopole) single magnetic monopole in universe would imply quantization of charge magnetic contributions can be ”rotated away “ as long asis constant(continutity)

Section 8.1 - Conservation of Charge and Energy* conservation of charge:* conservation of energy:equipotsurfflux tubePoynting vector”poynts “ in directionof energy flow(energy flux)wherefluxes:H-sheetsS-fluxtubes* Problem 8.2E-sheetseither real ordisplacement current

Section 8.2 - Conservation of Momentum* stress-energy tensor general relativity: electromagnetic fieldshave energy/momentumenergydensitycurvatureof spaceenergy fluxshearstressstress-energy(mass)pressuremomentum momentumdensityflux continuum mechanics:stress force / areastrain deformationdesity flux(wikipedia)energymomentum Newton’s law: stress causes the transfer of momentumconserved by Newton’s 3rd lawconservative forces* momentum theorema) pressure a) sheerP-wavesS-waves(diagonals)(off-diag)sheer stressforce / volumesame old story: convertusing Maxwellwant a full time derivative:divergence ofmomentum fluxif(linear material)electrodynamicstress tensor force / area momentum fluxconservation of momentumsymmetric:trace:

Section 9.1 - Wave Equation* enough already with the string equation* 3-d waves - another application of stressand straintensorHooke’s lawStrain displacemntNewton’s lawstrain displacement(Wikipedia)Elastic moduli (homogeneous, isotropic):Lamé’s 1st param ( pres.)Lamé’s 2nd param (sheer)Poisson’s ratioP-wave (longitudinal) modulusS-Wave (sheer,trans) modulusYoung’s modulusBulk ModulusNOTE: no relation whatsoever to H-atom S (l 0) and P (l 1) -waves* P-wave:* S-wave:for fluidsair:v 343 m/s @20degCwater: v 1482 m/ssteel: v 5960 m/s* mode-conversion: Zoeppritz eq’s (mech. equivalent of Frenel eq’s)

Solutions to the Wave Equation* separation of variables to form Helmholtz equation(phase velocity)d’AlembertianHelmholtz Eq. (wave eq. in frequency domain)* rectangular coords plane wave - now can be oscillating in all three dimensions, vs. Laplace:* cylindrical coordsletinteger order Bessel functionssolutions toLaplace eq’nfrom Wolfram Mathworld(Cylindrical) Bessel FunctionsSpherical Bessel Functions* spherical coordsspherical harmonics(associated Legendre fn’s)spherical Bessel equationspherical Bessel functions (1/2 integer order)solutions toLaplace eq’n

Section 9.1.2 - Harmonic Oscillations* Complex amplitudephase shiftphase”phasor“ complex amplitude, including phase shiftphase at a point in space, time note:is usually ommited in expression - real part is assumed addition of phasors: normal superpossition of wavesdo not mix multiplication of phasors: ?twice the frequency!just the magnitude, frequency drops out

* Example: LRC circuit(Serway) resonance phase shift: ELI the ICE manVoltage (E) in an Inductor (L) leads Current (I)Current (I) in a Capacitor (C) leads Voltage (E)

Section 9.1.3 - Reflection and Transmission* wave equation on a string* solution* energy densityconst* boundary conditionsconst(no point mass)* wave incident on a singularityincidentreflectedtransmitted* reflection and transmission coefficients* conservation of energy* phase shift transmitted wave always in phase reflected wave in phase if v2 v1; otherwise 180 degrees out of phase

Section 9.2 & 10.1 - Electromagnetic Waves in Vacuum* 10.1 Maxwell equations in terms of fields and potentials in the Lorenz gauge, where plane wave solution wave equation with 299792458 m/s in vacuum (def’n of the metre) other constants:definition of the Amperewhere do all the 9’s come from?* 9.2 sourceless wave equation in terms of fields plane waves:complex vector amplitude Maxwell’s equations are local in k-spaceform a RHS plane waves are featureless - no real direction of propagationphase velocity energy and momentumgroup velocity

Section 9.3 - Fresnel Equations* waves in matter - only difference: in terms of E, D, B, H, the difference is hidden in intensity* same boundary conditions* orthogonal incidence: only E,H conditions* oblique incidence: depends on polarizationP-polarization worked out here plane of incidence defined by geometrical opticsreflectionSnell’s law (refraction)

* oblique boundary conditions Fresnel equations

Section 9.4.1 - E&M Waves in Conductors* dissipation of free charge - RC time constant* Maxwell’s equations with sources & conductancespace-timefrequency domain* wave equation* complex square root* complex wave vector - wavelength and skin depth

Section 9.4.2 - Reflection & Transmission for Conductors* redo section 9.3. for conductors* mirror with finite but large conductivity

Section 9.4.3 - Dispersion in Nonconductive Media* damped harmonic oscillatorslow (infinite spring)resonance(sping and mass balanced)fast (infinite mass)* Breit-Wigner distribution* compare: LRC circuit

* complex index of refraction - damped harmonic oscillator model(wavelength, velocity) i (damping)absorption coefficient* off-resonance limit

Section 9.5.1 - Wave Guides* Maxwell’s equations* Wave guides - transverse-longitudinal separation microwaves, coaxial cable, transmission lines, linear accelerator cavities, fiber optics let wave propogate down the guide in the z-direction* it is enough to calculate the z-components* Waveguide modes: TEM mode- most similar to free plane waves, same spped of light, no cutoff frequency- cannot propagate in a hollow guide TE (transverse electric) TM (transverse magnetic

Section 9.5.2 - Rectangular Waveguide* wave equation for longitudinal component, boundary conditionsinside the conductor* general solution a superposition of:(TE)and(TM)* rectangular guide, TE mode: separate variables* frequency modes* cutoff frequency, lowest allowed frequency (k 0) wave vector not just from k, but also* phase v

potential ”flow“ surfaces divergent field ”flux“lines D from source Q V 1 V 2 I 5 I 4 I 3 I 2 I 1 I 0 I 6 discontinuity I 6 ! curling flow surfaces of H around source I closed flux lines B free charge in conductor bound charge in dielectric *

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