CERN Accelerating science

334 Pages·2010·16.6 MB·English

ELECTROMAGNETIC THEORY
A.B.Lahanas
UniversityofAthens,PhysicsDepartment,NuclearandParticlePhysicsSection,Athens15771,Greece
Abstract
An introduction to Electromagnetic Theory is given with emphasis on wave
propagation phenomena infreespaceandinsidewaveguides. Wealsodiscuss
theradiationemittedbymovingelectriccharges,anissuewhichisparticularly
important inaccelerator physics.
1. INTRODUCTION
Thetopics thatwillbecovered inthislecture arethefollowing:
MaxwellEquations
Conservation ofEnergy -ThePoynting vector
(cid:15)
Propagation ofElectromagnetic (EM)Waves
(cid:15)
PowerAbsorption byConducting Surfaces
(cid:15)
Propagation ofEMWavesinWaveGuides
(cid:15)
Energy FlowandPowerLossesinWaveGuides
(cid:15)
Potentials -Radiation byMovingCharges
(cid:15)
(cid:15) Allmaterialcoveredinthislectureanddetailsofthecalculationsinvolvedcanbefoundinstandard
textbooks (see for instance [1, 2, 3] ). In writing this lecture I have benefited from lectures given in
previous CERNschools (see[4]).
2. MAXWELLEQUATIONS
WestartbyreviewingMaxwell’s equations :
Electric charges whose density is are the sources of the electric field . In the system this is
expressed byGauss’s law (cid:26) E~ MKSA
(1)
1
~ E~ = (cid:26) :
r(cid:1) (cid:15)0
Electric currents with density are the sources of the magnetic induction field . This is
expressed byAmpere’s law J~ = (cid:26) ~u B~
(2)
~ B~ = (cid:22)0J~:
Field lines of are closed. This is equivarlen(cid:2)t to the statement that there are no magnetic monopoles.
MathematicallBy~thisisexpressed bytheequation
(3)
~ B~ = 0 :
Theelectromotiveforcearoundaclosedcircuriti(cid:1)sproportional totherateofchangeoffluxofthefield
through thecircuit (Faraday’s law). Indifferential formthislawisexpressed bythefollowing formulBa~
@B~ (4)
~ E~ = :
r(cid:2) (cid:0) @t
Within material media having polarization and magnetization theabove lawsstill hold with
thefollowing replacements P~ M~
@P~ @E~
(cid:26) (cid:26) ~ P~ ; J~ J~+ ~ M~ + +(cid:15)0 :
) (cid:0) r(cid:1) ) r(cid:2) @t @t
1
That is to the true charge density we have to add the polarization charge density and to the true current
density we have to add the contributions of the magnetization current, the polarization current and the
displacement current introduced by Maxwell. In terms of the electric displacement and magnetic fields,
defined by and respectively, Maxwellequations canbebrought
intothefollowD~ing f(cid:15)o0rmE~ + P~ H~ (cid:22)10 B~ M~
(cid:17) (cid:17) (cid:0)
~ D~ = (cid:26)
r(cid:1)
@B~
~ E~ =
r(cid:2) (cid:0) @t
@D~
~ H~ = J~ +
r(cid:2) @t
~ B~ = 0
r(cid:1)
Insomematerials( Linearmedia)ithappensthat wherethequantities
arecalled the dielectric constant and magnetic perD~me=abil(cid:15)itEy~o;fB~the=me(cid:22)diHu~m;respectively.
(cid:15); (cid:22)
2.1 TheContinuityEquation
The electric charge is conserved. Actually we have never observed in the laboratory a violation of this
conservation law. Thisconservation lawisexpressed bythefollowing Continuity Equation
(5)
@(cid:26)
~ J~ + = 0
r(cid:1) @t
where isthechargedensityand isthecurrentdensity. ThisequationfollowsfromMaxwell
equatio(cid:26)ns anditisnotanindependJ~ent=hy(cid:26)po~uthesis.
The quantity represents the charge flowing out of surface per unit time ( this is
measured in AmperesSiJ~n thde~Ssystem MKSA ). If the charge density is time inSdependent then from the
(cid:1)
continuity equation it follows that . In this case we say that we have steady currents. For
R
the steady current case the integral~ J~ = 0over any closed surface vanishes. In order to see its
consequence, consider the case ofrasS(cid:1)urJ~faced~Scrosing Nwires carrying cuSrrents flowinginto
(cid:1)
(or out) a node surrounded by the surface. Then the vanishing of this closed sI1u;rIf2a;c:e::i;nItNegral results to
H
,whichisthewellknownKirchoff’s current conservation law.
I1+I2+:::+IN = 0
2.2 TheLorentzforce
The force acting on a charge , which is at rest within an electric field, is . Also the force
acting on a small wire elemeent , carrying electric current , which is plF~ace=d ineaE~magnetic field, is
. These two sugd~glest that for acharge movIing with velocity , the total force acting
oF~ni=tisI d~l B~ e u
(cid:2)
(6)
F~ = e( E~ + ~u B~ ):
ThisisthewellknownLorentzforce. Foracontinuouschar(cid:2)geandcurrentdistribution, and ,
itisconvenient to definethe force density ,that isforce per unit volume. Onaccoun(cid:26)t ofeqJ~. (6=)th(cid:26)is~uis
givenby f~
(7)
f~ = (cid:26)E~ + J~ B~ :
(cid:2)
3. CONSERVATIONOFENERGY-ThePoyntingvector
From the previous section it becomes evident that the rate of doing work on a unit volume of the distri-
bution isgivenby
(8)
f~ ~u = (cid:26)(E~ + ~u B~ ) ~u = J~ E~ :
(cid:0) (cid:1) (cid:0) (cid:2) (cid:1) (cid:0) (cid:1)
2