electromagnetic theory - indico€¦ · electromagnetic theory g. franchetti, gsi cern accelerator...

9/29/16

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ElectromagneticTheory

G.Franchetti,GSICERNAccelerator– SchoolBudapest,2-14/10/2016

3/10/16 G.Franchetti 1

MathematicsofEM


Fieldsare3dimensionalvectorsdependentoftheirspatialposition(anddependingontime)

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Products


Scalarproduct

~A

~B

~A · ~B = AB cos ✓

Vectorproduct

~A

~B

~A⇥ ~B = AB sin ✓v

~A⇥~B

✓ ✓

Thegradientoperator


~r = (@x

, @y

, @z

)

Isanoperatorthattransformspacedependentscalarinvector

Example:givenf(x, y, z)

~rf(x, y, z) = (@x

f, @

y

f, @

z

f)

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Divergence/Curlofavectorfield


~

A(x, y, z)

Divergenceofvectorfield

~

A(x, y, z)

(@y

A

z

� @

z

A

y

)x+

(@z

A

x

� @

x

A

z

)y+

(@x

A

y

� @

y

A

x

)z

~r⇥ ~

A(x, y, z) =

~r · ~A(x, y, z) =

@x

Ax

+ @y

Ay

+ @z

Az

Curlofvectorfield

Relations


~A⇥ ( ~B ⇥ ~C) = �( ~A · ~B) ~C + ~B( ~A · ~C)

~r⇥ (~r⇥ ~C) = �(~r · ~r) ~C + ~r(~r · ~C)

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FluxConcept


Examplewithwater

ab


Volumepersecond

dV

dt= ava

a

or

L

θ

dV

dt

= Lbvcos✓

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Flux


θ~E

d ~A

A

d�( ~E) = ~E · d ~A

Fluxthroughasurface


�( ~E) =

Z

S

~E · d ~A

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Fluxthroughaclosedsurface:Gausstheorem


Z

S

~E · d ~A =

Z

Vr · ~E dV

Anyvolumecanbedecomposedinsmallcubes

Fluxthroughaclosedsurface

Stokestheorem


x

y

Ey(D,0,0)

Ey(0,0,0)

I

�

~E · d~l = Ex

(0, 0, 0)�+ Ey

(�, 0, 0)�

� Ex

(0,�, 0)�� Ey

(0, 0, 0)�

I

�

~

E · d~l =✓�@E

x

@y

+@E

y

@x

◆�2

I

�z

~E · d~l = (~r⇥ ~E)z�2

�

�

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foranarbitrarysurface


I

�

~E · d~l =Z

S(~r⇥ ~E) · d ~A

Howitworks


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ElectricChargesandForces


+ -Twocharges

Experimentalfacts

+ ++ -

--

Coulomblaw


q1

q2r

~F

~F =1

4⇡✏0

q1q2r2

r

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Units


SystemSI

~Fq1r

~F =1

4⇡✏0

q1q2r2

r

Newton

Coulomb

Meters

✏0 = 8.8541⇥ 10�12

C2N-1 m-2

permettivity offreespace

Superpositionprinciple


q1

q2r

~F

~F =1

4⇡✏0

q1q2r2

r

q3

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ElectricField


q1

q2r

~Fq3

~E =~F

q1


~F

~F = q1 ~E

~E

q1

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Byknowingtheelectricfieldtheforceonachargeiscompletelyknown

Workdonealongapath


W = q

Z

�

~E · d~l�

workdonebythecharge

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Electricpotential


V (P ) = �Z P

1~E · d~l

ForconservativefieldV(P)doesnotdependonthepath!

Centralforcesareconservative

UNITS:Joule/Coulomb=Volt

Workdonealongapath


�

workdonebythecharge

A

B

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ElectricFieldßà ElectricPotential


E

x

= � @

@x

V (x, y, z)

Ey = � @

@y

V (x, y, z)

Ez = � @

@z

V (x, y, z)

Invectorialnotation

~E = �~rV

Electricpotentialbyonecharge


Takeoneparticlelocatedattheorigin,then

V (~r) = �Z ~r

1

1

4⇡✏0

q

(r1 � r0)3(~r1 � ~r0) · d~l

V (~r) =1

4⇡✏0

q

r

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ElectricPotentialofanarbitrarydistribution


setofNparticles

q1

q2

q3

~r1

~r2

~r3

~rorigin

V (~r)

V (~r) =1

4⇡✏0

X

i

qip(~r � ~ri)2

Electricpotentialofacontinuousdistribution


Splitthecontinuousdistributioninagrid

qi = ⇢(~ri)dV

~ri

origin

V (~r) =1

4⇡✏0

X

i

qip(~r � ~ri)2

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Energyofachargedistribution


itistheworknecessarytobringthechargedistributionfrominfinity

Moresimply U =X

j

qjX

i=1,j

1

4⇡✏0

qi|~rj � ~ri|

Moresimply


Inintegralform

Using ~E = �~rV andthedivergencetheoremitcanbeprovedthat

✏0E2

2isthedensityofenergyoftheelectricfield

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Fluxoftheelectricfield


θ~E

d ~A

A

d�( ~E) = ~E · d ~A

Fluxofelectricfieldthroughasurface


�( ~E) =

Z

S

~E · d ~A

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ApplicationtoCoulomblaw

Z

S

~E · d ~A =q

✏0

Onasphere

Thisresultisgeneralandapplytoanyclosedsurface

Onanarbitraryclosedcurve


q1

q2

Z

S

~E · d ~A =1

✏0(q1 + q2)

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FirstMaxwellLaw


Z

S

~E · d ~A =Q

✏0integralform

forainfinitesimalsmallvolume differentialform

(trytoderiveit.Hint:usedGausstheorem)

~r · ~E =⇢

✏0

Physicalmeaning


Ifthereisachargeinoneplace,theelectricfluxisdifferentthanzero

Onechargecreateanelectricflux.

�( ~E) =q

✏0

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PoissonandLaplaceEquations


As ~E = �~rV and

combiningbothwefind

r2V = � ⇢

✏0

Invacuum: r2V = 0

~r · ~E =⇢

✏0

~r · ~rV = � ⇢

✏0

Poisson

Laplace

MagneticField


Thereexistnotamagneticcharge!(FindamagneticmonopoleandyougettheNobelPrize)

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Ampere’sexperiment


I1

I2

F = µ0I1I22⇡r

Ampere’sLaw


I

~B =µ0

2⇡

I

rv

I2

F

allexperimentalresultsfitwiththislaw

~F = I2d~l ⇥ ~B

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Units


~B =µ0

2⇡

I

rv

~F = dl~I2 ⇥ ~BFromN

Am= T [Tesla]

From

µ0 = 4⇡ ⇥ 10�7 N A�2

follows

Tohave1Tat10cmwithonecable I=5x105 Amperes!!

Biot-SavartLaw


I

~B =µ0

2⇡

I

rv

~r

d~l

d ~B =µ0

4⇡

d~l ⇥ ~r

r3

Fromanalogywiththeelectricfield

dB

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Lorentzforce


~F = q~v ⇥ ~B

Achargenotinmotiondoesnotexperienceaforce!

v

B

F

Noaccelerationusingmagneticfield!

+


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Fluxofmagneticfield


Thereexistnotamagneticcharge!Nomatterwhatyoudo..

Themagneticfluxisalwayszero!Z

S

~B · d ~A = 0

SecondMaxwellLaw


r · ~B = 0

Integralform

Differentialfrom

Z

S

~B · d ~A = 0

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ChangingthemagneticFlux...


~E = ~v ⇥ ~B

h

L

Magneticflux

Followingthepath

�( ~B) = hLB

E =1

h

d�( ~B)

dt

Z

�

~E · d~l = �d�( ~B)

dt

Faraday’sLaw


integralform

validinanywaythemagneticfluxischanged!!!

Z

�

~E · d~l = �d�( ~B)

dt

(Reallynotobvious!!)

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foranarbitrarysurface


I

�

~E · d~l =Z

S(~r⇥ ~E) · d ~A

Faraday’sLawindifferentialform


Z

S(~r⇥ ~E) · d ~A = � d

dt

Z

S

~B · d ~A

~r⇥ ~E = �@ ~B

@t

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SummaryFaraday’sLaw


Integralform

differentialform ~r⇥ ~E = �@ ~B

@t

I

�

~E · d~l = �d�( ~B)

dt

Importantconsequence


Currentcreatesmagneticfield

magneticfieldcreatemagneticflux

�(B) = LI

ChangingthemagneticfluxcreatesaninducedemfI

�

~E · d~l = �d�( ~B)

dt

L=inductance[Henry]

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I = 0

U =1

2LI2energynecessarytocreatethemagneticfield

h

r

✏emf = �d�(B)

dtdU = ✏emfIdt = �d�(B)

dtIdt


B = µ0NI

�(B) = V olume

B

2

µ0I

Therefore

V olume

B

2

µ0I= LI

U =1

2LI

2 = V olume

B

2

2µ0

B2

2µ0

Energydensityofthemagneticfield

Fieldinsidethesolenoid

Magneticflux �(B) = ⇡r2BNh

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Ampere’sLaw


I

B

I

�

~B · d~l = µ0I

DisplacementCurrent


I

�

~B · d~l = µ0I

I

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DisplacementCurrent


I

�

~B · d~l = µ0IHerethereisavaryingelectricfieldbutnocurrent!

I

DisplacementCurrent


StationarycurrentIà electricfieldchangeswithtime

I = ✏0@

@t

Z

S

~E · d ~A

I

ThisdisplacementcurrenthastobeaddedintheAmperelaw

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FinalformoftheAmperelaw


I

�

~B · d~l = µ0

✓I + ✏0

@

@t

Z

S

~E · d ~A◆

~r⇥ ~B = µ0

✓~j + ✏0

@

@t~E

◆

integralform

differentialform

MaxwellEquationsinvacuum


Integralform DifferentialformZ

S

~E · d ~A =Q

✏0r · ~E =

⇢

✏0Z

S

~B · d ~A = 0 r · ~B = 0

~r⇥ ~E = �@ ~B

@tI

�

~B · d~l = µ0

✓I + ✏0

@

@t

Z

S

~E · d ~A◆

~r⇥ ~B = µ0

✓~j + ✏0

@

@t~E

◆

I

�

~E · d~l = �d�( ~B)

dt

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Magneticpotential?


Canwefinda“potential”suchthat ~B = �~rV

~r · ~B = �r2VMaxwellequation

~r · ~B = 0

r2V = 0

~r⇥ ~B = ~r⇥ ~rV = 0But itmeansthatwecannotincludecurrents!!

?

Example:2Dmultipoles


For2Dstaticmagneticfieldinvacuum(onlyBx,By)

~B = �~rV~B = ~r⇥ ~A

�@x

V = @y

Az

@y

V = @x

Az

~B = (@y

Az

,�@x

Az

, 0)

~B = (�@x

V,�@y

V, 0)

ThesearetheCauchy-ReimannThatmakesthefunction

A+ iVanalytic

By

+ iBx

= �@x

(A+ iV )

By

+ iBx

= BX

n

(bn

+ ian

)zn

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VectorPotential


Ingeneralwerequire ~B = ~r⇥ ~A

~r · ~A = 0 (thischoiceisalwayspossible)

Automatically ~r · ~B = 0~r⇥ ~B = µ0

~J r2 ~A = �µ0~J

Solution


r2 ~A = �µ0~Jr2V = � ⇢

✏0

V (~r) =1

4⇡✏0

Z⇢(~ri)

|~r � ~ri|dV ~A =

µ0

4⇡

Z ~J(~r0)

|~r0 � ~r|dV

Electricpotential Magneticpotential

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Effectofmatter


Electricfield Magneticfield

ConductorsDielectric

DiamagnetismParamagnetismFerrimagnetism


Maxwellequationinvacuumarealwaysvalid,evenwhenweconsidertheeffectofmatter

Microscopicfield

Thatisthefieldis“local”betweenatomsandmovingcharges

Averagedfield

thisisafieldaveragedoveravolumethatcontainmanyatomsormolecules

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Conductors


freeelectrons boundedtobeinsidetheconductor

Conductorsandelectricfield


surfacedistributionofelectrons

boundedtobeinsidetheconductor

onthesurfacetheelectricfieldisalwaysperpendicular

~Eext

~E = 0

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�

� = ✏0EApplyingGausstheorem

�( ~E) = AE

Boundarycondition


Thesurfacesofmetalsarealwaysequipotential

++

+

- -

--

++ +

+ +

-

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Ohm’sLaw


freeelectrons

R =l

A⇢

l

A

⇢

[Ω]

resistivity

~E = ⇢ ~Jor

~J = � ~E

� =1

⇢conductivity

[Ωm]


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electromagnetic theory - indico€¦ · electromagnetic theory g. franchetti, gsi cern accelerator...

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