Short Review/Refresher Classical Electrodynamics (CED)

Transcription

1 Short Review/Refresher Classical Electrodynamics (CED) (.. and applications to accelerators) Werner Herr, CERN

2 Recommended Reading Material (in this order) [1 ] R.P. Feynman, Feynman lectures on Physics, Vol2. [2 ] Proceedings of CAS: RF f or accelerators, Ebeltoft, Denmark, 8-17 June 2010, Edited by R. Bailey, CERN [3 ] J.D. Jackson, Classical Electrodynamics (Wiley, ) [4 ] L. Landau, E. Lifschitz, The Classical Theory of Fields, Vol2. (Butterworth-Heinemann, 1975) [5 ] J. Slater, N. Frank, Electromagnetism, (McGraw-Hill, 1947, and Dover Books, 1970) Some refresher on required vector calculus in backup slides (Gauss, Stoke..)

3 OUTLINE This does not replace a full course (i.e. 60 hours, some additional material in backup slides, details in bibliography) Also, it cannot be treated systematically without special relativity. The main topics discussed: Basic electromagnetic phenomena Maxwell s equations Lorentz force and motion of particles in electromagnetic fields Electromagnetic waves in vacuum Electromagnetic waves in conducting media, waves in RF cavities and wave guides

4 Variables and units used in this lecture Formulae use SI units throughout. E( r, t) = electric field [V/m] H( r, t) = magnetic field [A/m] D( r,t) = electric displacement [C/m 2 ] B( r, t) = magnetic flux density [T] q = electric charge [C] ρ( r,t) = electric charge density [C/m 3 ] I, j( r,t) = current [A], current density [A/m 2 ] µ 0 = permeability of vacuum, 4 π 10 7 [H/m or N/A 2 ] ǫ 0 = permittivity of vacuum, [F/m] To save typing and space where possible (e.g. equal arguments): E( r,t) E same for other variables..

5 - ELECTROSTATICS -

6 Gauss theorem in the simplest form: Surface S enclosing a volume V within which are charges: q 1,q 2,... E E E E Electric field lines diverging from the enclosed charges.. E q +q E Sum up the fields passing through the surface flux Φ Φ = E n da = q i = Q S ǫ 0 ǫ 0 i n is the normal unit vector and E the electric field at an area element da of the surface Surface integral of E equals total charge Q inside enclosed volume

7 Φ = q ε. Φ = 0 Φ =.. 4q ε.. Essence: This holds for any arbitrary (closed) surface S, and: Does not matter how the particles are distributed inside the volume Does not matter whether the particles are moving Does not matter whether the particles are in vacuum or material

8 If we have not discrete charges but a continuous ) distribution: Replace charge by charge density q i volume dv. ρ = charge per unit For a charge density it is replaced by a volume integral: ρ E n da = dv = Q S V ǫ 0 ǫ }{{ 0 } this part is trivial The volume V is the one enclosed by the surface S ) obviously does not exist...

9 With some vector calculus: E da = ρ dv = Q S V ǫ 0 ǫ 0 = Φ E E da = E dv (relates surface and volume integrals) S } V {{ } Gauss formula E = ρ ǫ 0 written as divergence : div E Flux of electric field E through any closed surface is proportional to net electric charge Q enclosed in the region (Gauss Theorem). Written with charge density ρ we get Maxwell s first equation: div E = E = E x x + E y y + E z z = ρ ǫ 0 Divergence: measures outward flux Φ E of the field...

10 Simplest possible example: flux from a charge q A charge q generates a field E according to (Coulomb): E = q r 4πǫ 0 r 3 Enclose it by a sphere: E = const. on a sphere (area is 4π r 2 ): E da = q da = q 4πǫ 0 r 2 ǫ 0 sphere sphere Surface integral through sphere A is charge inside the sphere (any radius)

11 We can derive the field E from a scalar electrostatic potential φ(x,y,z), i.e.: then we have E = grad φ = φ = ( φ x, φ y, φ z ) E = 2 φ = ( 2 φ x + 2 φ 2 y + 2 φ ρ(x,y,z) 2 z2) = ǫ 0 This is Poisson s equation All we need is to find φ Example

12 Simplest possible charge distribution: point charge φ(r) = q 4πǫ 0 r E = φ(r) = q 4πǫ 0 r r 3

13 A very important example: 3D Gaussian distribution ρ(x,y,z) = (σ x,σ y,σ z r.m.s. sizes) Q σ x σ y σ z 2π 3 exp ( x2 2σ 2 x y2 2σ 2 y z2 2σ 2 z ) φ(x,y,z,σ x,σ y,σ z ) = Q 4πǫ 0 0 exp( x2 y2 z2 2σx 2 +t 2σy 2 +t 2σz 2+t) (2σx 2 +t)(2σy 2 +t)(2σz 2 +t) dt For the interested: Fields given in the backup slides For a derivation, see e.g. W. Herr, Beam-Beam Effects, in Proceedings CAS Zeuthen, 2003, CERN , and references therein.

14 Very important in practice: Poisson s equation in Polar coordinates (r, ϕ) 1 r r ( r φ ) r + 1 r 2 2 φ ϕ 2 = ρ ǫ 0 Poisson s equation in Cylindrical coordinates (r, ϕ, z) 1 r r ( r φ ) r + 1 r 2 2 φ ϕ φ z 2 = ρ ǫ 0 Poisson s equation in Spherical coordinates (r, θ, ϕ) 1 r 2 r ( r 2 φ ) r + 1 r 2 sinθ θ ( sinθ φ ) θ + 1 r 2 sinθ 2 φ ϕ 2 = ρ ǫ 0 Examples for solutions in [3]

15 - MAGNETOSTATICS -

16 Definitions: Magnetic field lines from North to South Properties: Described as vector fields All field lines are closed lines

17 Gauss second law... S B d A = V B dv = 0 B = 0 Closed field lines of magnetic flux density ( B): What goes out ANY closed surface also goes in, Maxwell s second equation: B = µ 0 H = 0 Physical significance: no Magnetic Charges (Monopoles)

18 From Ampere/Oersted law, for example current density j: Static electric current induces encircling (curling) magnetic field curl B = B = µ 0 j or in integral form the current density becomes the current I: B da = µ A A 0j da = µ 0I Curl: measures directional strength along the field lines...

19 Application (derivation see [1-5]): For a static electric current I in a single wire we get Biot-Savart law (we have used Stoke s theorem and area of a circle A = r 2 π): B = µ 0 4π B = µ 0 2π I r d s I r r 3 For magnetic field calculations in wires..

20 - THIS IS NOT THE WHOLE STORY - - enter Maxwell -

21 Do we need an electric current? Maxwell s displacement current, e.g. a charging capacitor j d : Defining a Displacement Current I d : I d = dq dt = ǫ 0 dφ dt = ǫ 0 d dt Not a current from moving charges area E d A But a current from time varying electric fields

22 Displacement current I d produces magnetic field, just like actual currents do... Time varying electric field induce magnetic field (using the current density j d B = µ 0 j d = ǫ 0 µ 0 E t

23 Bottom line: Magnetic fields B can be generated in two ways: B = µ 0 j (electric current, Ampere) B = µ 0 j d = ǫ 0 µ 0 E t (changing electric field, Maxwell) or putting them together: B = µ 0 ( j + j d ) = µ 0 j +ǫ 0 µ 0 E t or as integral equations (using Stoke s formula): B d r = B da = ( E µ A 0 j +ǫ 0 µ 0 C A t }{{} Stoke s formula ) d A

24 - enter Faraday - - first unification -

25 Faraday s law (electromagnetic induction): v I v I S N S N I I A changing flux Ω through an area A produces electromotive force (EMF) in a conducting coil: current I (Can move magnet or coil: any relative motion will do..) flux = Ω = Bd A EMF = E d s Ω t A = Bd A t = A }{{} flux Ω C E d s C

26 Ω t = A t Bd A = C E d s In a conducting coil: changing flux induces circulating current Flux can be changed by: - Change of magnetic field B with time t (e.g. transformers) - Change of area A with time t (e.g. dynamos) Electromotive force (EMF): 1. Energy of a unit charge after one loop 2. Voltage if the loop is cut, i.e. open circuit

27 A B t d A = E da = E d s A } {{ C } Stoke sformula enclosed area (A) E. ds E = B t closed curve (C) Changing field through any closed area induces electric field in the (arbitrary) boundary becomes Maxwell-Faraday law

28 Summary: Time Varying Fields (most significant for RF systems!) db d t d E d t E B Time varying magnetic fields produce curling electric field: curl( E) = E = d B t Time varying electric fields produce curling magnetic field: curl( B) = B = µ 0 ǫ 0 d E t because of the they are perpendicular: E B

29 Put together: Maxwell s Equations in vacuum (SI units) E = ρ ǫ 0 = φ B = 0 E = d B ( dt B d E = µ 0 j + ǫ 0 dt ) (I) (II) (III) (IV) (a.k.a. Microscopic Maxwell equations)

30 For completeness A E da = Q ǫ 0 B da = 0 A E d s = ( ) d B da C A dt ( B C d d s = µ 0 j +ǫ ) E A 0 dt d A Equivalent equations written in Integral Form, (using Gauss and Stoke s formulae)

31 Maxwell in Physical terms 1. Electric fields E are generated by charges and proportional to total charge 2. Magnetic monopoles do not exist 3. Changing magnetic flux generates circumscribing electric fields/currents 4.1 Changing electric flux generates circumscribing magnetic fields 4.2 Static electric current generates circumscribing magnetic fields Frequent complaint: I have seen them in a different form! The Babel of Units:

32 Units: Gauss law Ampere/Maxwell SI E = ρ ǫ 0 B = µ 0 j + µ 0 ǫ 0 d E dt Electro-static (ǫ 0 = 1) E = 4πρ B = 4π c 2 j + 1 d E c 2 dt Electro-magnetic (µ 0 = 1) E = 4πc 2 ρ B = 4π j + 1 c 2 d E dt Gauss cgs E = 4πρ B = 4π c j + 1 c Lorentz E = ρ B = 1 c j + 1 c d E dt d E dt Also: B Gauss = 4π µ 0 B SI ρ Gauss = ρsi 4πǫ0 and so on...

33 That s not all Electromagnetic fields in material In vacuum: D = ǫ 0 E, B = µ0 H In a material: D = ǫ r ǫ 0 E = ǫ 0 E + P B = µ r µ 0 H = µ 0 H + M Origin: P olarization and M agnetization ǫ r ( E, r,ω) ǫ r is relative permittivity [ ] µ r ( H, r,ω) µ r is relative permeability [0(!) 10 6 ] (i.e.: linear, isotropic, non-dispersive)

34 Once more: Maxwell s Equations D = ρ B = 0 E = d B dt H = j + d D dt (a.k.a. Macroscopic Maxwell equations)

35 Something on potentials (needed in lecture on Relativity): Electric fields can be written using a (scalar) potential φ: E = φ Since div B = 0, we can write B using a (vector) potential A: B = A = curl A combining Maxwell(I) + Maxwell(III): E = φ A t Fields can be written as derivatives of scalar and vector potentials Φ(x,y,z) and A(x,y,z) (absolute values of potentials Φ and A can not be measured..)

36 The Coulomb potential of a static charge q is written as: Φ( r) = 1 4πǫ 0 q r r q where r is the observation point and r q the location of the charge The vector potential is linked to the current j: 2 A = µ 0 j The knowledge of the potentials allows the computation of the fields see lecture on relativity (fields of moving charges)

37 Applications of Maxwell s Equations Powering of magnets Lorentz force, motion in EM fields - Motion in electric fields - Motion in magnetic fields EM waves (in vacuum and in material) Boundary conditions EM waves in cavities and wave guides

38 Powering and self-induction - Induced magnetic flux B changes with changing current Induces a current and magnetic field B i voltage in the conductor Induced current will oppose change of current (Lenz s law) We want to change a current to ramp a magnet...

39 Ramp rate defines required Voltage: U = L I t Inductance L in Henry (H) Example: - Required ramp rate: 10 A/s - With L = 15.1 H per powering sector Required Voltage is 150 V

40 Lorentz force on charged particles Moving ( v) charged (q) particles in electric ( E) and magnetic ( B) fields experience a force f (Lorentz force): f = q ( E + v B) Why a mysterious and incomprehensible dependence on the velocity of the charge??? Often treated as ad hoc plugin to Maxwell s equation, but it is not (see lecture on Special Relativity )!!

41 Motion in an electric field E F.q The solution is: d dt (m 0 v) = f = q E v = q E m 0 t r = q E 2m 0 t 2 (parabola) Constant E-field deflects beams: TV, electrostatic separators (SPS,LEP)

42 Motion in magnetic fields Assume first no electric field: d dt (m 0 v) = f = q v B Force is perpendicular to both, v and B No forces on particles at rest Why: see lecture on special relativity

43 Important application: Tracks from particle collisions, lower energy particles have smaller bending radius, allows determination of momenta.. Q1: what is the direction of the magnetic field??? Q2: what is the charge of the incoming particle???

44 Example: Motion in a magnetic dipole Practical units: B[T] ρ[m] = p[ev/c] c[m/s] Example LHC: B = 8.33 T, p = 7000 GeV/c ρ = 2804 m

45 Use of static fields (some examples, incomplete) Magnetic fields Bending magnets Focusing magnets (quadrupoles) Correction magnets (sextupoles, octupoles, orbit correctors,..) Electric fields Electrostatic separators (beam separation in particle-antiparticle colliders) Very low energy machines What about non-static, time-varying fields?

46 Time Varying Fields (very schematic) B(t) E(t).... Time varying magnetic fields produce circular electric fields Time varying electric fields produce circular magnetic fields Can produce self-sustaining, propagating fields (i.e. waves) Example for source (classical picture): oscillating charge

47 Electromagnetic waves (classical picture) Vacuum: only fields, no charges (ρ = 0), no current (j = 0)... From: E = B t = ( E) = ( B t ) = ( 2 E) = t ( B) = ( 2 E) = µ0 ǫ 0 2 E t 2 It happens to be: µ 0 ǫ 0 = 1 c 2 2 E 1 2 E = c 2 t = µ 2 0 ǫ 0 2 E and B B = t 2 c 2 t = µ 2 0 ǫ 0 2 B t 2 General form of a wave equation

48 Solutions of the wave equations: E = E 0 e i( k r ωt) B = B 0 e i( k r ωt) k = 2π λ = ω c (propagation vector) λ = (wave length, 1 cycle) ω = (frequency 2π) c = ω = (wave velocity) k Magnetic and electric fields are transverse to direction of propagation: E B k k E0 = ω B 0 Speed of wave in vacuum: c = m/s

49 Examples: Spectrum of EM waves (we are exposed to) Radio as low as 40 Hz ( ev ) CMB Hz ( 10 3 ev ) yellow light Hz ( 2 ev ) X rays Hz ( 4 kev ) γ rays Hz ( 12 MeV ) π 0 γγ Hz ( 70 MeV )

50 Polarization of EM waves (Classical Picture!): The solutions of the wave equations imply monochromatic plane waves: E = E 0 e i( k r ωt) B = B0 e i( k r ωt) Look now only at electric field, re-written using unit vectors in the plane transverse to propagation: ǫ 1 ǫ 2 k Two Components: E1 = ǫ 1 E 1 e i( k r ωt) E2 = ǫ 2 E 2 e i( k r ωt) E = ( E 1 + E 2 ) = ( ǫ 1 E 1 + ǫ 2 E 2 ) e i( k r ωt) With a phase shift φ between the two directions: E = ǫ 1 E 1 e i( k r ωt) + ǫ 2 E 2 e i( k r ωt+φ) φ = 0: linearly polarized light φ 0: elliptically polarized light φ = ± π 2 and E 1 = E 2 : circularly polarized light

51 Polarized light - why interesting: Produced (amongst others) in Synchrotron light machines (linearly and circularly polarized light, adjustable) blue sky! Accelerator and other applications: Polarized light reacts differently with charged particles Beam diagnostics, medical diagnostics (blood sugar,..) Inverse FEL Material science 3-D motion pictures, LCD display, outdoor activities, cameras (glare),......

52 Energy in electromagnetic waves (in brief, details in [2, 3, 4]): We define as the Poynting vector (SI units): S = 1 µ 0 E B (in direction of propagation) describes the energy flux, i.e. energy crossing a unit area, per J second [ m 2 s ] In free space: energy in a plane is shared between electric and magnetic field The energy density H would be: H = 1 ) (ǫ 0 E 2 + 1µ0 B 2 2

53 Waves interacting with material Need to look at the behaviour of electromagnetic fields at boundaries between different materials (air-glass, air-water, vacuum-metal,...). Have to consider two particular cases: Ideal conductor (i.e. no resistance), apply to: - RF cavities - Wave guides Conductor with finite resistance, apply to: - Penetration and attenuation of fields in material (skin depth) - Impedance calculations Can be derived from Maxwell s equations, here only the results!

54 Observation: between air and water Some of the light is reflected Some of the light is transmitted and refracted Reason are boundary conditions for fields between two materials

55 Extreme case: surface of ideal conductor For an ideal conductor (i.e. no resistance) the tangential electric field must vanish Corresponding conditions for normal magnetic fields. We must have: E t = 0, Bn = 0 This implies: Fields at any point in the conductor are zero. Only some field patterns are allowed in waveguides and RF cavities A very nice lecture in R.P.Feynman, Vol. II Now for Boundary Conditions between two different regions

56 Boundary conditions for electric fields Material 1 Material 2 Material 1 Material 2 ε 1 µ 1 ε 2 µ 2 ε 1 µ 1 ε 2 µ 2 E t D n Assuming no surface charges (proof e.g. [3, 5]) : From curl E = 0: tangential E-field continuous across boundary (E 1 t = Et) 2 From div D = ρ: normal D-field continuous across boundary (D 1 n = Dn) 2 with surface charges, see backup slides

57 Boundary conditions for magnetic fields Material 1 Material 2 ε 1 µ 1 ε 2 µ 2 Material 1 Material 2 ε 1 µ 1 ε 2 µ 2 H t B n Assuming no surface currents (proof e.g. [3, 5]) : From curl H = j: tangential H-field continuous across boundary (H 1 t = Ht) 2 From div B = 0: normal B-field continuous across boundary (B 1 n = Bn) 2 with surface current, see backup slides

58 Summary: boundary conditions for fields Electromagnetic fields at boundaries between different materials with different permittivity and permeability (ǫ 1,ǫ 2,µ 1,µ 2 ). (Et 1 = Et), 2 (En 1 En) 2 (Dt 1 Dt), 2 (Dn 1 = Dn) 2 (Ht 1 = Ht), 2 (Hn 1 Hn) 2 (Bt 1 Bt), 2 (Bn 1 = Bn) 2 (derivation deserves its own lectures, just accept it) They determine: reflection, refraction and refraction index n.

59 Reflection and refraction angles related to the refraction index n and n : incident wave x reflected wave sin α sin β = n n = tan α B n = εµ α z n depends on wave length dn dλ < 0 n = ε µ β refracted wave If light is incident under angle α B [3]: Reflected light is linearly polarized perpendicular to plane of incidence (Application: fishing air-water gives α B 53 o )

60 Rectangular cavities and wave guides Rectangular, conducting cavities and wave guides (schematic) with dimensions a b c and a b: x x a a b z b z y c y Fields must be zero at boundary RF cavity, fields can persist and be stored (reflection!) Plane waves can propagate along wave guides, here in z-direction

61 Assume a rectangular RF cavity (a, b, c), ideal conductor. Without derivations (e.g. [2, 3, 6]), the components of the fields are: E x = E x0 cos(k x x) sin(k y y) sin(k z z) e iωt E y = E y0 sin(k x x) cos(k y y) sin(k z z) e iωt E z = E z0 sin(k x x) sin(k y y) cos(k z z) e iωt with: E = B t : B x = i ω (E y0k z E z0 k y ) sin(k x x) cos(k y y) cos(k z z) e iωt B y = i ω (E z0k x E x0 k z ) cos(k x x) sin(k y y) cos(k z z) e iωt B z = i ω (E x0k y E y0 k x ) cos(k x x) cos(k y y) sin(k z z) e iωt

62 Consequences for RF cavities No fields outside: field must be zero at conductor boundary! Only possible under the condition: and for k x,k y,k z we can write: k 2 x +k 2 y +k 2 z = ω2 c 2 k x = m xπ a, k y = m yπ b, k z = m zπ c, The integer numbers m x,m y,m z are called mode numbers number of half-wave patterns across width and height It means that a half wave length λ/2 must always fit exactly the size of the cavity.

63 Allowed modes Modes in cavities Allowed Allowed Not allowed a Only modes which fit into the cavity are allowed λ 2 = a 4, λ 2 = a 1, λ 2 = a 0.8 No electric field at boundaries, wave must have nodes at the boundaries

64 Similar considerations lead to (propagating) solutions in (rectangular) wave guides: E x = E x0 cos(k x x) sin(k y y) e i(k zz ωt) E y = E y0 sin(k x x) cos(k y y) e i(k zz ωt) E z = i E z0 sin(k x x) sin(k y y) e i(k zz ωt) B x = 1 ω (E y0k z E z0 k y ) sin(k x x) cos(k y y) e i(k zz ωt) B y = 1 ω (E z0k x E x0 k z ) cos(k x x) sin(k y y) e i(k zz ωt) B z = 1 i ω (E x0k y E y0 k x ) cos(k x x) cos(k y y) e i(k zz ωt)

65 Consequences for wave guides Similar considerations as for cavities, no field at boundary. We must satisfy again the condition: k 2 x +k 2 y +k 2 z = ω2 c 2 This leads to modes like (no boundaries in direction of propagation z): k x = m xπ a, k y = m yπ b, The numbers m x,m y are called mode numbers for planar waves in wave guides!

66 Re-writing the condition as: k 2 z = ω2 c 2 k2 x k 2 y k z = Propagation without losses requires k z to be real, i.e.: ω 2 c > 2 k2 x +ky 2 = ( m xπ a )2 +( m yπ ) 2 b ω 2 c 2 k2 x k 2 y which defines a cut-off frequency ω c. For lowest order mode: ω c = π c a Above cut-off frequency: propagation without loss At cut-off frequency: standing wave Below cut-off frequency: attenuated wave (means it does not really fit and k is complex).

67 Classification of wave guide modes: TE: no E-field in z-direction TM: no B-field in z-direction TEM: no B-field nor E-field in z-direction What is special: TEM modes cannot propagate in a single conductor )! Need two concentric conducting cylinders : i.e. a coaxial cable... (for the field lines: see backup slides) ) curl E = 0, div E = 0, E = 0 at boundaries zero field

68 Circular cavities Wave guides and cavities are more likely to be circular. Derivation using the Laplace equation in cylindrical coordinates, example for modes, for the derivation see e.g. [2, 3]: E r E θ E z = E 0 k z k r J n(k r ) cos(nθ) sin(k z z) e iωt = E 0 nk z k 2 rr J n(k r ) sin(nθ) sin(k z z) e iωt = E 0 J n (k r r) cos(nθ) sin(k z z) e iωt B r = ie 0 ω c 2 k 2 rr J n(k r r) sin(nθ) cos(k z z) e iωt B θ = ie 0 ω c 2 k r r J n(k r r) cos(nθ) cos(k z z) e iωt B z = 0 Homework: write it down for wave guides..

69 Accelerating circular cavities For accelerating cavities we need longitudinal electric field component E z 0 and magnetic field purely transverse. E r = 0 E θ = 0 E z = E 0 J 0 (p 01 r R ) e iωt B r = 0 B θ = i E 0 c J r 1(p 01 B z = 0 R ) e iωt (p nm is the mth zero of J n, e.g. p ) This would be a cavity with a TM 010 mode: ω 010 = p 01 c R

70 Other case: finite conductivity Starting from Maxwell equation: Wave equations: H = j + d D dt E = E 0 e i( k r ωt), = j {}}{} σ {{ E } + ǫ d E dt Ohm s law H = H0 e i( k r ωt) We want to know k, applying the calculus to the wave equations we have: de = iω E, dt Put together: d H dt = iω H, E = i k E, H = i k H k H = iσ E ωǫ E = ( iσ +ωǫ) E

71 Starting from: With B = µ H: k H = iσ E + ωǫ E = ( iσ +ωǫ) E E = i k E = B t = µ H t = iωµ H Multiplication with k: k ( k E) = ωµ( k H) = ωµ( iσ +ωǫ) E After some calculus and E H k: k 2 = ωµ( iσ + ωǫ)

72 Skin Depth Using k 2 = ωµ( iσ + ωǫ): For a good conductor σ ωǫ: k 2 iωµσ k ωµσ 2 (1+i) = 1 δ (1+i) δ = 2 ωµσ is the Skin Depth High frequency currents avoid penetrating into a conductor, flow near the surface (Note: i = e iπ/4 = [(1+i)/2] 2)

73 Explanation - inside a conductor (very schematic) I e I e I H I e I e eddy currents from changing H-field: Cancel current flow in the centre of the conductor Enforce current flow at the skin (surface) Q: Why are high frequency cables thin??

74 Attenuated waves - penetration depth Waves incident on conducting material are attenuated Is basically Skin depth, (attenuation to 1/e) Wave form: e i(kz ωt) = e i((1+i)z/δ ωt) = e z δ e i(z δ ωt)

75 Examples and applications Skin Depth versus frequency Copper Gold Stainless steel Carbon (amorphous) skin depth (m) e-06 1e-08 1e e+06 1e+08 1e+10 1e+12 1e+14 1e+16 Frequency (Hz) Skin depth Copper: 1 GHz: δ 2.1 µm, 50 Hz: δ 10 mm (there is an easy way to waste your money...) Penetration depth Seawater: to get δ 25 m you need 76 Hz inefficient ( ) and very low bandwidth (0.03 bps)

76 Skin Depth - beam dynamics For metal walls thicker than δ: Resistive Wall Impedances, see later on collective effects. Currents penetrate into the wall, depending on the frequency and conductivity. For the transverse impedance we get the dependence: Z t (ω) δ ω 1/2 Largest impedance at low frequencies Cause instabilities (see later)

77 We are done... Review of basics and Maxwell s equations Lorentz force Motion of particles in electromagnetic fields Electromagnetic waves in vacuum Electromagnetic waves in media Waves in RF cavities Waves in wave guides Penetration of waves in material

78 However...! Have to deal with moving charges! Electromagnetic wave concept fuzzy: no medium! Lorentz force depends on frame of reference! Mutual interactions between charges and fields! Cannot explain details of Cherenkov and Transition Radiation To sort it out in a systematic framework (but ignoring Quantum Effects): Special Relativity...

79 - BACKUP SLIDES -

80 Boundary conditions in the presence of surface charges and currents Assuming surface charges σ s and currents j s, we get the boundary conditions: µ 1H (1) n = µ 2H (2) n ǫ 1E (1) n ǫ 2E (2) n = σ s D (1) t D = (2) t B (1) t B (2) t = j s ǫ 1 ǫ 2 µ 1 µ 2 Another assumption: both media are linear and isotropic, i.e. B = µ H D = ǫ E

81 Coaxial cable: H field Poynting vector x x x x x x x x E field Field lines and Poynting vector in a coaxial cable

82 Side notes: Remark 1: B often written curl B or rot B (mostly Europe) Remark 2: On very few occasions one can see it written as: B = µ 0 j Sometimes used in France, but usually it refers to a different algebra. If interested, see backup slides for the meaning and relevance, happy reading..

83 Vector calculus... We can define a special vector (sometimes written as ): = ( x, y, z ) It is called the gradient and invokes partial derivatives. It can operate on a scalar function φ(x,y,z): φ = ( φ x, φ y, φ z ) = G = (G x,g y,g z ) and we get a vector G. It is a kind of slope (steepness..) in the 3 directions. Example: φ(x,y,z) = C ln(r 2 ) with r = x 2 +y 2 +z 2 φ = (G x,g y,g z ) = ( 2C x r 2, 2C y r 2, 2C z r 2 )

84 Gradient (slope) of a scalar field Lines of pressure (isobars) Gradient is large (steep) where lines are close (fast change of pressure)

85 Vector calculus... The gradient can be used as scalar or vector product with a vector F, sometimes written as Used as: F or F Same definition for products as before, treated like a normal vector, but results depends on how they are applied: Φ is a vector F is a scalar F is a pseudo-vector F is not a vector

86 What about the operation? In general dimensions: - No analogue of a cross product to yield a vector - The product is not a normal vector, but a 2-vector (or bi-vector) - Can be interpreted as a normal cross product by mapping 2-vectors to normal vectors by using the Hodge dual: a b = (a b) (aha...)

87 Operations on vector fields... Two operations of have special names: Divergence (scalar product of gradient with a vector): div( F) = F = F 1 x + F 2 y + F 3 z Physical significance: amount of density, (see later) Curl (vector product of gradient with a vector): ( curl( F) = F F3 = y F 2 z, F 1 z F 3 x, F 2 x F ) 1 y Physical significance: amount of rotation, (see later)

88 Meaning of Divergence of fields... Field lines of a vector field F seen from some origin: F < 0 F > 0 F = 0 (sink) (source) (fluid) The divergence (scalar, a single number) characterizes what comes from (or goes to) the origin

89 How much comes out? Surface integrals: integrate field vectors passing (perpendicular) through a surface S (or area A), we obtain the Flux: F da A Density of field lines through the surface (e.g. amount of heat passing through a surface)

90 Surface integrals made easier... Gauss Theorem: Integral through a closed surface (flux) is integral of divergence in the enclosed volume F da = F dv A V F closed surface (A) da enclosed volume (V) Relates surface integral to divergence

91 Meaning of curl of fields The curl quantifies a rotation of vectors: 8 2D vector field 6 2D vector field y 0 y x x Line integrals: integrate field vectors along a line C: F d r sum up vectors (length) in direction of line C (e.g. work performed along a path...) C

92 Line integrals made easier... Stokes Theorem: Integral along a closed line is integral of curl in the enclosed area F d s = F da C A enclosed area (A) E. ds closed curve (C) Relates line integral to curl

93 Two vector fields: Integration of (vector-) fields 8 2D vector field 6 2D vector field y 0 y x x F = 0 F 0 F 0 F = 0 C F d r = A F d A Line integral for second vector field vanishes...

94 Scalar products Define a scalar product for (usual) vectors like: a b, a = (x a,y a,z a ) b = (xb,y b,z b ) a b = (x a,y a,z a ) (x b,y b,z b ) = (x a x b +y a y b + z a z b ) This product of two vectors is a scalar (number) not a vector. (on that account: Scalar Product) Example: ( 2,2,1) (2,4,3) = = 7

95 Vector products (sometimes cross product) Define a vector product for (usual) vectors like: a b, a = (x a,y a,z a ) b = (xb,y b,z b ) a b = (x a,y a,z a ) (x b,y b,z b ) = (y a z b z a y }{{} b, z a x b x a z b, x }{{} a y b y a x b ) }{{} x ab y ab z ab This product of two vectors is a vector, not a scalar (number), (on that account: Vector Product) Example 1: ( 2,2,1) (2,4,3) = (2,8, 12) Example 2 (two components only in the x y plane): ( 2,2,0) (2,4,0) = (0,0, 12)

96 Is that the full truth? If we have a circulating E-field along the circle of radius R? should get acceleration! Remember Maxwell s third equation: E d r = d dt C 2πRE θ A = dφ dt B d A

97 Motion in magnetic fields This is the principle of a Betatron Time varying magnetic field creates circular electric field! Time varying magnetic field deflects the charge! For a constant radius we need: m v2 R = e v B B = p e R t B(r,t) = 1 dp e R dt B(r,t) = πr 2 BdS B-field on orbit must be half the average over the circle condition Betatron

98 Fields from Gaussian distribution - 2D Φ(x,y,σ x,σ y ) = Q e ( x 2 y2 2σx 2 +t 2σy 2+t) 4πǫ 0 0 (2σ 2 x +t)(2σy 2 +t) dt E x = E y = [ ( Q 2ǫ 0 2π(σ 2 x σy) Im w 2 [ ( Q 2ǫ 0 2π(σ 2 x σy) Re w 2 ) x+iy 2(σ 2 x σy) 2 ) x+iy 2(σ 2 x σy) 2 ( e x 2 2σx 2 + y2 σ x y 2σy 2 w ( e x 2 2σx 2 + y2 σ x y 2σy 2 w σ x +iy σ x σ y 2(σ 2 x σ 2 y) σ x +iy σ x σ y 2(σ 2 x σ 2 y) )] )] here w(z) is the complex error function From: M. Basetti and G. Erskine, CERN-ISR-TH/80-06