Class3:Electromagnetism

Inthisclasswewillapplyindexnotationtothefamiliarfieldofelectromagnetism,anddiscuss

itsdeepconnectionwithrelativity

Class3:Electromagnetism

Attheendofthissessionyoushouldbeableto…

• … describehowelectromagnetismisfundamentallyarelativisticphenomenon

• … understandhowelectromagnetismcanbeformulatedinindexnotationusingtheMaxwellFieldTensor

• … usetheLorentztransformationstotransformelectricandmagneticfieldsbetweenframes

• … describeelectromagneticenergydensityandflowintermsofindexnotation

Synthesisofideas

• Physicsisaboutthesynthesisofideas:understandingapparentlydifferentphenomenaasthejointconsequencesofadeeperreality

Forexample,in1666NewtonrealizedthatthesamegravitationalforcecausesbothapplestofalldownwardsontheEarth,andtheEarthtoorbittheSun

https://www.pinterest.co.uk/pin/477451997975217238https://www.space.com/16080-solar-system-planets.html

Synthesisofideas

• Electromagnetism representsanothergreatsynthesisofideas.Priorto1830,electricity andmagnetism wereconsideredseparatephenomena

http://www.miniscience.com/kits/KITSEC/index.html https://www.dreamstime.com/stock-image-magnet-nails-image29533081

Electricity Magnetism

• However,Faraday’sexperimentsdemonstratedthatamagnet,aswellasabattery,candriveacurrent

https://www.electronics-tutorials.ws/electromagnetism/electromagnetic-induction.html

Electricityandmagnetismareconnectedbydeeperprinciples!

Synthesisofideas

Electricfield

• Electricchargegeneratesanelectricfield𝐸 inthesurroundingregion

• Anelectricfieldisaregionofspaceinwhichanelectricchargefeelsaforce,�⃗� = 𝑞𝐸

https://en.wikipedia.org/wiki/Electric_field

• Electriccurrent(ormovingcharge)generatesamagneticfield𝐵 inthesurroundingregion

• Amagneticfieldisaregionofspaceinwhichanelectriccurrentfeelsaforce,�⃗� = 𝐼𝑑𝑙×𝐵

Magneticfield

• Maxwell’sEquationsdescribehow𝐸 and𝐵-fieldsaregeneratedfromchargeandcurrent

http://www.hotelsrate.org/maxwell-equation-t-shirt/http://izquotes.com/quote/277247

Maxwell’sEquations

• Potentials:𝐸 and𝐵-fieldscanbegeneratedfromanelectrostaticpotential𝑉 andmagneticpotential𝐴

• Electrostatics:𝛻×𝐸 = 0 → 𝑬 = −𝜵𝑽

• Magnetism:𝛻. 𝐵 = 0 → 𝑩 = 𝜵×𝑨

• Time-varying:𝛻×𝐸 + 9:9;= 0 → 𝑬 = −𝜵𝑽 − 𝝏𝑨

𝝏𝒕

Electromagneticpotentials

https://www.pinterest.com.au/pin/512143788857285159/

• Energy:ittakesworktoestablishan𝐸 or𝐵-field

• Wemaythinkofthisworkasbeingstoredinthefields withenergydensity>

?𝜀A𝐸? or

>?BC

𝐵?

• Thefluxofenergyisgivenby >BCD×:E

(the“Poynting vector”)

https://www.miniphysics.com/ss-magnetic-field-due-to-current-in-a-solenoid.htmlhttps://www.quora.com/Why-doesnt-a-dielectric-change-the-electric-field-generated-by-a-capacitor-V-constant

Electromagneticenergy

4-vectorsforelectromagnetism

• Since𝐸 isproducedbycharges,and𝐵 bymovingcharges,it’snaturalthat𝐸 and𝐵 arelinkedbyviewingindifferentreferenceframes – i.e.,byaLorentztransformation

• Whenalineofchargemoves,itbecomesacurrent!

𝑣

𝑆 𝑆′

𝑥 𝑥′

𝜆 = chargedensity

4-vectorsforelectromagnetism

• Since𝐸 isproducedbycharges,and𝐵 bymovingcharges,it’snaturalthat𝐸 and𝐵 arelinkedbyviewingindifferentreferenceframes – i.e.,byaLorentztransformation

• However,todevelopelectromagnetisminrelativisticnotation,weneedsome4-vectors!

• Inthelastclasswesawthatthesourcesofthefieldsforma4-vector,𝐽B = (𝜌𝑐, 𝐽P, 𝐽Q, 𝐽R) withcontinuity𝜕B𝐽B = 0

• Inthisclasswewilldemonstratethatelectromagnetismmaybedescribedverynicelyifthepotentialsalsoforma4-vector,𝐴B = (U

E, 𝐴P, 𝐴Q, 𝐴R)

Maxwellfieldtensor

• We’llformawonderfulobjectknownastheMaxwellFieldTensor (“tensor”becauseithas2indices,𝜇 and𝜈)

• “Ifindoubt,writeitout”

𝐹BX = 𝜕B𝐴X − 𝜕X𝐴B

𝜕A𝐴A − 𝜕A𝐴A 𝜕A𝐴> − 𝜕>𝐴A𝜕>𝐴A − 𝜕A𝐴> 𝜕>𝐴> − 𝜕>𝐴>

𝜕A𝐴? − 𝜕?𝐴A 𝜕A𝐴Y − 𝜕Y𝐴A𝜕>𝐴? − 𝜕?𝐴> 𝜕>𝐴Y − 𝜕Y𝐴>

𝜕?𝐴A − 𝜕A𝐴? 𝜕?𝐴> − 𝜕>𝐴?𝜕Y𝐴A − 𝜕A𝐴Y 𝜕Y𝐴> − 𝜕>𝐴Y

𝜕?𝐴? − 𝜕?𝐴? 𝜕?𝐴Y − 𝜕Y𝐴?𝜕Y𝐴? − 𝜕?𝐴Y 𝜕Y𝐴Y − 𝜕Y𝐴Y

𝜕B = −1𝑐𝜕𝜕𝑡 ,

𝜕𝜕𝑥 ,

𝜕𝜕𝑦 ,

𝜕𝜕𝑧

𝐴B =𝑉𝑐 , 𝐴P, 𝐴Q, 𝐴R

𝜇 = 0𝜇 = 1𝜇 = 2𝜇 = 3

𝜈 = 0 𝜈 = 1 𝜈 = 2 𝜈 = 3

Maxwellfieldtensor

• We’llformawonderfulobjectknownastheMaxwellFieldTensor (“tensor”becauseithas2indices,𝜇 and𝜈)

• Multiplyingthisout,

𝐹BX = 𝜕B𝐴X − 𝜕X𝐴B

𝐹BX =

0 𝐸P/𝑐−𝐸P/𝑐 0

𝐸Q/𝑐 𝐸R/𝑐𝐵R −𝐵Q

−𝐸Q/𝑐 −𝐵R−𝐸R/𝑐 𝐵Q

0 𝐵P−𝐵P 0

𝜕B = −1𝑐𝜕𝜕𝑡 ,

𝜕𝜕𝑥 ,

𝜕𝜕𝑦 ,

𝜕𝜕𝑧

𝐴B =𝑉𝑐 , 𝐴P, 𝐴Q, 𝐴R

Maxwell’sequations

Maxwell’sequations…

… canbenicelywrittenas

𝛻. 𝐸 =𝜌𝜀A

𝛻×𝐸 +𝜕𝐵𝜕𝑡 = 0

𝛻. 𝐵 = 0

𝛻×𝐵 − 𝜀A𝜇A𝜕𝐸𝜕𝑡 = 𝜇A𝐽

𝜕B𝐹BX = −𝜇A𝐽X

𝜕a𝐹BX + 𝜕B𝐹Xa + 𝜕X𝐹aB = 0

TransformationofEMfields

• WecanusetheMaxwellFieldTensortodeducehowelectromagneticfieldstransformbetweenframes:

𝐹′BX = 𝐿Bc𝐿Xa𝐹ca

𝐿BX =

𝛾 −𝑣𝛾/𝑐−𝑣𝛾/𝑐 𝛾

0 00 0

0 00 0

1 00 1

𝐹BX =

0 𝐸P/𝑐−𝐸P/𝑐 0

𝐸Q/𝑐 𝐸R/𝑐𝐵R −𝐵Q

−𝐸Q/𝑐 −𝐵R−𝐸R/𝑐 𝐵Q

0 𝐵P−𝐵P 0

Lorentztransformation

Maxwellfieldtensor

TransformationofEMfields

• Multiplyingouteachcomponentofthisequation:

• 𝐸 and𝐵 “intermingle”underaLorentztransformation!

𝐸P′ = 𝐸P𝐸Q′ = 𝛾 𝐸Q − 𝑣𝐵R𝐸R′ = 𝛾 𝐸R + 𝑣𝐵Q

𝐵P′ = 𝐵P𝐵Q′ = 𝛾 𝐵Q + 𝑣𝐸R/𝑐?

𝐵R′ = 𝛾 𝐵R − 𝑣𝐸Q/𝑐?

Anotherpathtorelativity?

• Maxwell’sEquationskeepthesamemathematicalformunderaLorentztransformation– butnotunderaGalileantransformation

• ThiswasalreadyknownbyLorentzinthe1890s,beforeEinsteinpublishedSpecialRelativity

• InthemodernstoryEinstein’spostulatesarepresentedfirst,butitcanallbederivedfromelectromagnetism! H.Lorentz (1853-1928)

https://www.nobelprize.org/nobel_prizes/physics/laureates/1902/lorentz-bio.html

Magneticfieldofacurrent

• Whenalineofchargemoves,itbecomesacurrent

• Firstconsiderthefieldofthestationarychargein𝑆′:

𝑣

𝑆 𝑆′

𝑥 𝑥′

𝐸′

Gauss’sLaw:

𝐸Qe =𝜆′

2𝜋𝜀A𝑦′

𝜆 = chargedensity

𝐵e = 0

Magneticfieldofacurrent

• Whatisthemagneticfieldin𝑆?

• Lorentztransformation:𝐵R =ghDij

Ek= ghaj

?lmCQjEk

• Lengthcontractionimpliesthatthelinedensityofchargein𝑆 is𝛾𝜆′,andthecurrentin𝑆 isthen𝐼 = 𝛾𝜆e𝑣

• Hence,using𝑦e = 𝑦 and𝑐? = >mCBC

,wefind𝐵R =BCn?lQ

• Ampere’sLaw!

https://www.quora.com/Does-a-current-carrying-conductor-act-as-a-magnet-Does-it-have-to-be-wound-around-an-iron-core-to-get-the-magnetic-effect

𝑥

𝑦

⨀𝑧

TransformationofEMfields

• Whataretheimplicationsofthisformulationofelectromagnetism?

• Magnetismisnotanindependentphenomenon,butarelativisticeffect thatoccurswhenelectrostaticsareobservedinadifferentframe

• Ifthespeedoflightwereinfinite,therewouldbenomagnetism!

Fieldofamovingcharge

• Whatistheelectricfieldofacharge+𝑞 movingthrough𝑆?

• Firstconsiderthefieldofastationarychargein𝑆e:

+𝒒 𝑣

𝑆 𝑆′

𝑥 𝑥′

https://courses.lumenlearning.com/boundless-physics/chapter/the-electric-field-revisited/

𝐸e =𝑞

4𝜋𝜀A𝑟?�̂�′ 𝐵′ = 0

Fieldofamovingcharge

• Lorentztransformations:𝐸 = 𝐸P′, 𝛾𝐸Q′, 𝛾𝐸R′

• Evaluatethefieldat𝑡 = 0:𝑟e = 𝛾𝑥, 𝑦, 𝑧

• Aftersomealgebra:𝑟′? = 𝛾?𝑟? 1 − 𝑣 sin 𝜃 /𝑐 ?

+𝒒 𝑣𝜃

𝑃𝐸

𝐸 =𝑞𝑟

4𝜋𝜀A𝛾?𝑟Y 1 − 𝑣 sin 𝜃 /𝑐 ? Y/?

EMenergy-momentumtensor

• Theenergy-momentumtensorforelectromagnetismis:

𝑇BX =1𝜇A

𝐹Ba𝐹Xa −14 𝜂

BX𝐹ca𝐹ca

• Howdowede-codethisexpression??

• 𝐹Ba = 𝜂aX𝐹BX = ∑ 𝜂aX𝐹BX�X

• 𝐹ca = 𝜂cB𝜂aX𝐹BX

• 𝐹ca𝐹ca = ∑ ∑ 𝐹ca𝐹ca�a

�c

𝜂BX =−1 00 1

0 00 0

0 00 0

1 00 1

𝐹BX =

0 𝐸P/𝑐−𝐸P/𝑐 0

𝐸Q/𝑐 𝐸R/𝑐𝐵R −𝐵Q

−𝐸Q/𝑐 −𝐵R−𝐸R/𝑐 𝐵Q

0 𝐵P−𝐵P 0

• Theenergydensityisgivenbythefirstcomponent,𝑇AA:

• Evaluatingthisexpression,𝑇AA = >?𝜀A𝐸? +

>?BC

𝐵?

• Thefluxofenergyinthe𝑖-directionisgivenby𝑇A~:

• Evaluatingthisexpression,𝑇A~ = >BC

D×:E

𝑇AA =1𝜇A

𝐹Aa𝐹Aa −14 𝜂

AA𝐹ca𝐹ca

EMenergy-momentumtensor

𝑇A~ =1𝜇A

𝐹Aa𝐹~a −14 𝜂

A~𝐹ca𝐹ca

= 0

Forceonamovingcharge

• TheLorentzforcelawcanbeexpressedinindexnotationas

𝑑𝑝B

𝑑𝜏 = 𝑞𝐹BX𝑢X

Equivalentto�⃗� = 𝑞 𝐸 + �⃗�×𝐵

Gauges

• Inelectrostatics,thezeroofpotentialisarbitrary

• Ingeneral,if𝐴B → 𝐴B + 𝜕B𝑓,where𝑓 isanyfunctionof�⃗�, 𝑡 ,theelectromagneticfields𝐸 and𝐵 areunchanged!(Wecanshowthisbysubstitutingin𝐹BX = 𝜕B𝐴X − 𝜕X𝐴B)

• Thisfreedomisknownasagaugechoice– it’sconvenienttochoose𝑓 suchthat𝜕B𝐴B = 0

• That’sbecausewethenfind,𝜕B𝐹BX = 𝜕B𝜕B𝐴X

• CombiningwithMaxwell’sEquation𝜕B𝐹BX = −𝜇A𝐽X:

−1𝑐?

𝜕?

𝜕𝑡? +𝜕?

𝜕𝑥? +𝜕?

𝜕𝑦? +𝜕?

𝜕𝑧? 𝐴B = −𝜇A𝐽B