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

Transcription

1 lectromagnetism Christopher R Prior Fellow and Tutor in Mathematics Trinity College, Oxford ASTeC Intense Beams Group Rutherford Appleton Laboratory

2 Contents Review of Maxwell s equations and Lorentz Force Law Motion of a charged particle under constant lectromagnetic fields Relativistic transformations of fields lectromagnetic waves Waves in vacuo Waves in conducting medium Waves in a uniform conducting guide Simple example T 1 mode Propagation constant, cut-off frequency Group velocity, phase velocity Illustrations

3 Reading J.D. Jackson: Classical lectrodynamics H.D. Young and R.A. Freedman: University Physics (with Modern Physics) P.C. Clemmow: lectromagnetic Theory Feynmann Lectures on Physics W.K.H. Panofsky and M.N. Phillips: Classical lectricity and Magnetism G.L. Pollack and D.R. Stump: lectromagnetism 3

4 Basic quations from Vector Calculus For a scalar function ϕ ( x,y,z,t), gradient : ϕ ϕ x, ϕ y, ϕ z For a vector F ( F 1,F,F 3 ) divergence : F F 1 x + F y + F 3 z curl : F F 3 y F z, F 1 z F 3 x, F x F 1 y Gradient is normal to surfaces φconstant 4

5 Basic Vector Calculus ( F G) G F F ϕ, F ( F) ( F) F G S Stokes Theorem F ds F dr ds Oriented boundary C n ds C n Divergence or Gauss Theorem F dv V S F ds Closed surface S, volume V, outward pointing normal 5

6 What is lectromagnetism? The study of Maxwell s equations, devised in 1863 to represent the relationships between electric and magnetic fields in the presence of electric charges and currents, whether steady or rapidly fluctuating, in a vacuum or in matter. The equations represent one of the most elegant and concise way to describe the fundamentals of electricity and magnetism. They pull together in a consistent way earlier results known from the work of Gauss, Faraday, Ampère, Biot, Savart and others. Remarkably, Maxwell s equations are perfectly consistent with the transformations of special relativity.

7 Maxwell s quations Relate lectric and Magnetic fields generated by charge and current distributions. electric field D electric displacement H magnetic field B magnetic flux density ρ charge density j current density µ (permeability of free space) 4π 1-7 ε (permittivity of free space) c (speed of light) m/s In vacuum D ε, B µ H, ε µ c 1 D ρ B B t H j + D t

8 ρ ε Maxwell s 1 st quation quivalent to Gauss Flux Theorem: ρ ε V dv The flux of electric field out of a closed region is proportional to the total electric charge Q enclosed within the surface. A point charge q generates an electric field sphere ds q 4πε S q 4πε r ds 3 r sphere ds r 1 ε q ε V ρ dv Area integral gives a measure of the net charge enclosed; divergence of the electric field gives the density of the sources. 8 Q ε

9 B Maxwell s nd quation B Gauss law for magnetism: B d S The net magnetic flux out of any closed surface is zero. Surround a magnetic dipole with a closed surface. The magnetic flux directed inward towards the south pole will equal the flux outward from the north pole. If there were a magnetic monopole source, this would give a non-zero integral. Gauss law for magnetism is then a statement that There are no magnetic monopoles

10 B t Maxwell s 3 rd quation quivalent to Faraday s Law of Induction: S C (for a fixed circuit C) ε dl The electromotive force round a circuit is proportional to the rate of change of flux of magnetic field, through the circuit. ds d l d dt S S B t B ds d S dφ dt Faraday s Law is the basis for electric generators. It also forms the basis for inductors and transformers. Φ B ds N S

11 B µ j + Originates from Ampère s (Circuital) Law : C 1 c t B d l S Maxwell s 4 th quation B d S µ S B µ j j d S µ I Ampère Biot Satisfied by the field for a steady line current (Biot-Savart Law, 18): µ I B 4π dl r 3 r For a straight line current B θ µ I π r

12 Need for Displacement Current Faraday: vary B-field, generate -field Maxwell: varying -field should then produce a B-field, but not covered by Ampère s Law. Apply Ampère to surface 1 (flat disk): line integral of B μ Surface 1 Surface I Applied to surface, line integral is zero since no current penetrates the deformed Current I surface. Closed loop B µ Q ε A In capacitor,, so dq dt I ε A Displacement current density is t ( j + j ) j d µ + µ ε t j d ε d dt 1

13 Consistency with Charge Conservation Charge conservation: Total current flowing out of a region equals the rate of decrease of charge within the volume. j ds d dt j dv j + ρ t ρdv ρ dv t From Maxwell s equations: Take divergence of (modified) Ampère s equation B µ j + 1 c t B µ j + 1 c µ j + ε µ t j + ρ t ρ ε ( ) t Charge conservation is implicit in Maxwell s quations 13

14 Maxwell s quations in Vacuum In vacuum D ε, B µ H, ε µ 1 c Source-free equations: B + B t Source equations: ρ ε B 1 c t µ j quivalent integral forms (useful for simple geometries) d S 1 ε ρdv B ds d l d dt B ds dφ dt B d l µ j ds + 1 d c dt ds 14

15 xample: Calculate from B z dl d dt B ds B z B sinωt Also from B t B µ j + 1 c r r r < > dt r r r r < > r r π r θ θ π r θ θ d π r B sinωt dt 1 Bω r cos ωt d π r dt ω r B r then gives current density necessary to sustain the fields B sinωt cosωt π r π r B ω cosωt B ω cosωt

16 Lorentz Force Law Thought of as a supplement to Maxwell s equations but actually implicit in relativistic formulation, gives force on a charged particle moving in an electromagnetic field: f q + v B For continuous distributions, use force density Relativistic equation of motion 4-vector form: F dp dτ ( ) f d ρ + j B γ ( v f c 3-vector component: nergy component:, f ) γ ( 1 c d dt, d p ) dt d dt ( m γ v ) ( ) f q + v B v f d dt m dγ c dt 16

17 Motion of charged particles in constant magnetic fields d dt d dt ( m γ v ) f q + v B 1. From energy equation, γ is constant No acceleration with a magnetic field. From momentum equation, B d dt γ v B v ( ) q ( m γ c ) v f qv v B ( ) γ d dt ( ) v constant and v constant v also constant v B v // is constant 17

18 Motion in Constant magnetic field d dt ( m γ v ) qv v ρ q m γ v B B d v dt q v B m γ Constant magnetic field gives uniform spiral about B with constant energy. circular motion with radius ρ m γ v qb at angular frequency ω v ρ qb m (m m γ ) m γ v q B ρ Magnetic p q rigidity

19 Motion in Constant lectric Field d dt ( m γ v ) d dt γ f q + v B q m Solution of ( v) is γ v q t γ 1 + γ v m c ( ) d dt ( m γ v ) q γ 1 + q m c t dx dt γ v γ nergy gain is qx x x + m c x + 1 q q m t 1 + qt m c 1 for q << m c Constant -field gives uniform acceleration in straight line 19

20 Relativistic Transformations of and B According to observer O in frame F, particle ( has ) velocity v, fields are and B and Lorentz force is f q + v B In Frame F, particle is at rest and force is Assume measurements give same charge and force, so Point charge q at rest in F: See a current in F, giving a field Suggests Rough idea f q q q and + v B B B 1 c v q v r 4πɛ r 3, B B µ q 4π v r r 3 1 c v

21 Review of Waves u 1D wave equation is with general x solution 1 u v t u(x, t) f(vt x)+g(vt + x) Simple plane wave: 1D : sin ωt k x ( ) ( ) 3D : sin ωt k x Wavelength is Frequency is λ π k ν ω π

22 Phase and group velocities Plane wave sin(ωt kx) has constant phase ωt kx 1π at peaks ω k x v p x t ω k Superposition of plane waves. While shape is relatively undistorted, pulse travels with the Group Velocity v g dω dk A( k) e [ ω( k t kx] i ) dk

23 Wave packet structure Phase velocities of individual plane waves making up the wave packet are different, The wave packet will then disperse with time 3

24 lectromagnetic waves Maxwell s equations predict the existence of electromagnetic waves, later discovered by Hertz. No charges, no currents: ( ) t B t ( B ) H D t D 3D wave equation: B t B µ D t µɛ t x + y + z µɛ t ( ) ( )!!

25 Nature of lectromagnetic Waves A general plane wave with angular frequency ω travelling in the direction of the wave vector Phase invariant. Apply Maxwell s equations: has the form exp[ i( ωt k x)] B B exp[ i( ωt k x)] ik t iω π number of waves and so is a Lorentz B B Waves are transverse to the direction of propagation, and are mutually perpendicular ω t k x, B k k k k B k ωb

26 Plane lectromagnetic Wave 6

27 Plane lectromagnetic Waves B Combined with deduce that B 1 ω k B c t c k ωb ω kc k ω speed of wave in vacuum is k ω c Wavelength Frequency π λ k ν ω π Reminder: The fact that invariant tells us that Λ ( ω c, k ωt k x ) is an is a Lorentz 4-vector, the 4-Frequency vector. Deduce frequency transforms as ω γ(ω v k) ω c v c + v

28 Waves in a Conducting Medium exp[ i( ωt k x)] B B exp[ i( ωt k x)] j σ (Ohm s Law) For a medium of conductivity σ, Modified Maxwell: Put Dissipation factor D σ ωɛ H j + ɛ t σ + ɛ t i k H σ + iωɛ conduction current displacement current Copper : Teflon : σ σ , ε ε, ε.1ε D D

29 Attenuation in a Good Conductor i k H σ + iωε k H iσ ωε Combine with B t k ( k ) ω µ k H ω µ iσ ωε k k k ω µ iσ ωε ( ) k ω µ iσ + ωε For a good conductor D >> 1, Wave form is where δ ωµσ k ω µ H ( ) ( ) ( ) since x x exp i ωt exp, δ δ k ωµσ σ >> ωε, k iωµσ k 1 is the skin - depth 1 k δ ( 1 i) ( i)

30 Maxwell s quations in a Uniform Perfectly Conducting Guide Hollow metallic cylinder with perfectly conducting boundary surfaces Maxwell s equations with time dependence exp(iωt) are: ( ) ( ) + 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 t + (ω εµ + γ ) [ ] H γ is the propagation constant Can solve for the fields completely in terms of z and H z z x y

31 A simple model: Parallel Plate Waveguide y x z x xa Transport between two infinite conducting plates (T 1 mode): (,1,) ( x) t d dx i.e. e ( iωt γ z) K, where ( x) satisfies K sin A Kx cos ω εµ + γ To satisfy boundary conditions, on x and xa, need Asin Kx, K K Propagation constant is n nπ, n integer a nπ ω γ K n ω εµ 1 ωc a ω where c K n εµ 31

32 Cut-off frequency, ω c γ nπ a 1 ω ω c, A sin nπx a eiω t γ z, ω c nπ a εµ ω<ω c gives real solution for γ, so attenuation only. No wave propagates: cut-off modes. ω>ω c gives purely imaginary solution for γ, and a wave propagates without attenuation. γ ik, k εµ ω ω c ( ) 1 ω εµ 1 ω c For a given frequency ω only a finite number of modes can propagate. ω 1 ω > ω c nπ a εµ n < aω π εµ For given frequency, convenient to choose a s.t. only n1 mode occurs. 3

33 Waveguide animations above cut-off lower ω at cut-off below cut-off variable ω 33

34 Phase and group velocities in the simple Wave number: Wavelength: Phase velocity: Group velocity: wave guide k εµ ( ω ω ) 1 c < ω εµ π π λ >, k ω εµ v p ω > k 1, εµ larger ( ) v g dω k εµ ω ω c the free space wavelength than free - space dk k ωεµ < 1 εµ smaller than free - space velocity velocity 34

35 Calculation of Wave Properties If a3 cm, cut-off frequency of lowest order mode is f c ωc π 1 a εµ GHz At 7 GHz, only the n1 mode propagates and ω c nπ a εµ k εµ ω ( ω c ) 1 π ( 7 5 ) / m 1 λ π k 6cm v p ω k ms 1 > c v g k ωεµ.1 18 ms 1 < c 35

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