Electromagnetism. Christopher R Prior. ASTeC Intense Beams Group Rutherford Appleton Laboratory

Transcription

1 letomagnetism Chistophe R Pio Fellow and Tuto in Mathematis Tinity College Oxfod ASTeC Intense Beams Goup Ruthefod Appleton Laboatoy

2 Contents Maxwell s equations and Loentz Foe Law Motion of a haged patile unde onstant letomagneti fields Relativisti tansfomations of fields letomagneti waves Waves in a unifom onduting guide Simple example T 1 mode Popagation onstant ut-off fequeny Goup veloity phase veloity Illustations

3 Reading J.D. Jakson: Classial letodynamis H.D. Young and R.A. Feedman: Univesity Physis (with Moden Physis) P.C. Clemmow: letomagneti Theoy Feynmann Letues on Physis W.K.H. Panofsky and M.N. Phillips: Classial letiity and Magnetism G.L. Pollak and D.R. Stump: letomagnetism 3

4 What is eletomagnetism? The study of Maxwell s equations devised in 1863 to epesent the elationships between eleti and magneti fields in the pesene of eleti hages and uents whethe steady o apidly flutuating in a vauum o in matte. The equations epesent one of the most elegant and onise way to desibe the fundamentals of eletiity and magnetism. They pull togethe in a onsistent way ealie esults known fom the wok of Gauss Faaday Ampèe Biot Savat and othes. Remakably Maxwell s equations ae pefetly onsistent with the tansfomations of speial elativity. 4

5 Maxwell s quations Relate leti and Magneti fields geneated by hage and uent distibutions. eleti field D eleti displaement H magneti field B magneti flux density ρ hage density j uent density μ (pemeability of fee spae) 4π 1-7 ε (pemittivity of fee spae) (speed of light) m/s In vauum D ρ B B D H j + D ε B μh ε μ 1 5

6 Maxwell s 1 st quation ρ quivalent to Gauss Flux Theoem: ε ρ ε V dv S ds 1 ε ρdv Q ε The flux of eleti field out of a losed egion is popotional to the total eleti hage Q enlosed within the sufae. A point hage q geneates an eleti field sphee ds q 4πε q 4πε 3 sphee ds q ε Aea integal gives a measue of the net hage enlosed; divegene of the eleti field gives the density of the soues. 6

7 Maxwell s nd quation B B S Gauss law fo magnetism: B d The net magneti flux out of any losed sufae is zeo. Suound a magneti dipole with a losed sufae. The magneti flux dieted inwad towads the south pole will equal the flux outwad fom the noth pole. If thee wee a magneti monopole soue this would give a non-zeo integal. Gauss law fo magnetism is then a statement that Thee ae no magneti monopoles 7

8 Maxwell s 3 d quation B quivalent to Faaday s Law of Indution: ds (fo a fixed iuit C) S C dl d dt S B ds S B ds dφ dt The eletomotive foe ound a iuit dl is popotional to the ate of hange of flux of magneti field Φ B ds though the iuit. ε N S Faaday s Law is the basis fo eleti geneatos. It also foms the basis fo indutos and tansfomes. 8

9 Maxwell s 4 th quation Oiginates fom Ampèe s (Ciuital) Law : B μ j + B dl B ds μ j ds μi C S S B μ j 1 Ampèe Satisfied by the field fo a steady line uent (Biot-Savat Law 18): Biot Fo μi dl B 3 4π a staight line uent B θ μi π 9

10 Need fo displaement uent Faaday: vay B-field geneate -field Maxwell: vaying -field should then podue a B-field but not oveed by Ampèe s Law. Apply Ampèe to sufae 1 (flat disk): line Sufae 1 Sufae integal of B μ I Applied to sufae line integal is zeo sine no uent penetates the defomed Cuent I sufae. Q dq d In apaito so I ε A ε A dt dt Closed loop Displaement uent density is j d ε B μ ( j + j ) μ j d + με 1

11 Consisteny with hage onsevation Chage onsevation: Total uent flowing out of a egion equals the ate of deease of hage within the volume. j ds j + j dv ρ d dt ρ dv ρ dv Fom Maxwell s equations: Take divegene of (modified) Ampèe s equation 1 B μ j + ρ μ j + t ε μ ε ρ j + ( ) Chage onsevation is impliit in Maxwell s s quations 11

12 Maxwell s quations in Vauum In vauum D ε B μ H ε μ Soue-fee equations: Soue equations B v B + v B ρ ε 1 μ j 1 quivalent integal foms (sometimes useful fo simple geometies) v ds B ds dl B dl μ 1 ε d dt dφ B ds dt 1 d j ds + ds dt ρ dv 1

13 xample: Calulate fom B z dl d dt B ds B z B Also fom sinωt < > B 1 B μ j + dt < > π θ π θ θ θ π B ω osωt Bω osωt π ω B B ω osωt osωt then gives uent density neessay to sustain the fields 13

14 Loentz foe law Supplement to Maxwell s equations gives foe on a haged patile moving in an eletomagneti field: ( f q + v B) Fo ontinuous distibutions have a foe density ρ + j f d B Relativisti equation of motion 4-veto fom: F dp dτ γ v f 1 f γ d dt dp dt 3-veto omponent: d dt γ ( m v ) f q ( + v B) 14

15 Motion of haged patiles in onstant eletomagneti fields Constant -field gives unifom aeleation in staight line Solution of x m q 1 q m negy gain t d dt 1+ fo q qt m << d dt ( m v) f q ( + v B) q m 1 m γ ( γ v) qx Constant magneti field gives unifom spial about B with onstant enegy. dv dt v// onstant B x onstant q v m γ 15

16 16 Relativisti Tansfomations of and B Aoding to obseve O in fame F patile has veloity fields ae and and Loentz foe is In Fame F patile is at est and foe is Assume measuements give same hage and foe so Point hage q at est in F: See a uent in F giving a field Suggests ( ) B v q f + v B q f B v q q + and 4 3 B q πε v v q B 3 1 4π μ v B B 1 ( ) // // // // B B v B B B v + γ γ xat:

17 letomagneti waves Maxwell s equations pedit the existene of eletomagneti waves late disoveed by Hetz. No hages no uents: ( ) D μ ( B) B με H D D B B ( ) ( ) 3D wave equation : + x y + z με 17

18 Natue of eletomagneti waves A geneal plane wave with angula fequeny ω tavelling in the dietion of the wave veto has the fom k exp[ i( ωt k x)] B B exp[ i( ωt ω t k x Phase π numbe of waves and so is a Loentz invaiant. Apply Maxwell s equations ik iω k x )] B k k & B k ω B Waves ae tansvese to the dietion of popagation and k ae mutually pependiula s B and B 18

19 Plane eletomagneti wave 19

20 Plane letomagneti Waves 1 ω B k B Combined with k ωb ω k dedue that B k ω speed of wave in vauum is k ω Wavelength Fequeny π λ k ν ω π ω t k x The fat that is an invaiant tells us that ω Λ k is a Loentz 4-veto the 4-Fequeny veto. Dedue fequeny tansfoms as ( ) v ω γ ω v k ω + v

21 Waves in a onduting medium Fo a medium of ondutivity σ Modified Maxwell: & & H j + ε σ + ε Put Dissipation fato D σ ωε ik H j σ + ondution uent σ exp[ i( ωt k x)] B B exp[ i( ωt k x)] iωε displaement uent Coppe : Teflon : σ σ ε ε ε.1ε D D

22 Attenuation in a Good Conduto Combine with k Fo a good onduto D >> 1 Wave whee fom is δ ik & B ( k ) ωμ k H ωμ ( iσ + ωε ) ω μσ k H ωμ σ >> ωε exp i ωt σ + iωε ( iσ + ωε ) x exp δ k ωμh ω k iωμσ k μσ 1 x δ is the skin - depth ( i) oppe.mov wate.mov

23 3 Maxwell s quations in a unifom pefetly onduting guide z y x Hollow metalli ylinde with pefetly onduting bounday sufaes Maxwell s equations with time dependene exp(jωt)ae: ( ) ( ) H H i i t D H H i t B με ω εμ ω ωμ ωε ωμ Assume ) ( ) ( ) ( ) ( ) ( ) ( z t i z t i e y x H t z y x H e y x t z y x γ ω γ ω Then [ ] ) ( + + H t γ εμ ω Can solve fo the fields ompletely in tems of z and H z γ is the popagation onstant

24 Speial ases Tansvese magneti (TM modes): H z eveywhee z on ylindial bounday Tansvese eleti (T modes): z eveywhee on ylindial bounday Tansvese eletomagneti (TM modes): z H z eveywhee equies γ n H z + ω εμ o γ ± iω εμ 4

25 5 A simple mode: Paallel Plate Waveguide z x y Tanspot between two infinite onduting plates (T 1 mode): x xa t ) ( satisfies ) ( whee ) ( (1) γ εμ ω γ ω + K K dx d x e x z t i Kx A os sin i.e. To satisfy bounday onditions on x and xa so intege sin n a n K K Kx A n π Popagation onstant is εμ ω ω ω π εμ ω γ n n K a n K whee 1

26 Cut-off fequeny ω γ nπ a ω 1 ω Asin nπx a e iωt γ z ω nπ a εμ ω<ω gives eal solution fo γ so attenuation only. No wave popagates: ut-off modes. ω>ω gives puely imaginay solution fo γ and a wave popagates without attenuation. 1 ω ( ) 1 γ ik k εμ ω ω ω εμ ω Fo a given fequeny ω only a finite numbe of modes an popagate. nπ aω ω > ω n < εμ a εμ π 1 Fo given fequeny onvenient to hoose a s.t. only n1 mode ous. 6

27 Popagated eletomagneti fields Fom B i H ωμ (assuming A is eal) Ak nπ x H x sin os ωμ a H y A nπ nπ x H z os sin ωμ a a ( ωt kz) ( ωt kz) x z 7

28 Phase and goup veloities Plane wave exp j(ωt-kx) has onstant phase ωt-kx at peaks ω Δ t kδ x v p Δ Δ x t ω k A( k ) e [ ω ( k t kx ] i ) dk Supeposition of plane waves. While shape is elatively undistoted pulse tavels with the goup veloity v g dω dk 8

29 Wave paket stutue Phase veloities of individual plane waves making up the wave paket ae diffeent The wave paket will then dispese with time 9

30 Phase and goup veloities in the simple wave guide Wave numbe is so wavelength in guide k 1 ( ω ω ) ω εμ εμ < the fee-spae wavelength Phase veloity is lage than fee-spae veloity v p ω > k π π λ > k ω εμ 1 εμ Goup veloity is less than infinite spae value k εμ ( ω ω ) v g dω k < dk ωεμ 1 εμ 3

31 Calulation of wave popeties If a3 m ut-off fequeny of lowest ode mode is f ω π a 1 εμ 5GHz At 7 GHz only the n1 mode popagates and k εμ ( ω ω ) 1 π λ 6m k ω 8 1 v p ms k k 8 vg.1 1 ms ωεμ 13m

32 Waveguide animations T1 mode above ut-off T1 mode smalle ω T1 mode at ut-off T1 mode below ut-off T1 mode vaiable ω T mode above ut-off T mode smalle T mode at ut-off T mode below ut-off ppwg_1-1.mov ppwg_1-.mov ppwg_1-3.mov ppwg_1-4.mov ppwg_1_vf.mov ppwg_-1.mov ppwg_-.mov ppwg_-3.mov ppwg_-4.mov 3