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Electromagnetism.ppt
- 2. Contents • Review of Maxwell’s equations and Lorentz Force Law • Motion of a charged particle under constant Electromagnetic fields • Relativistic transformations of fields • Electromagnetic energy conservation • Electromagnetic waves – Waves in vacuo – Waves in conducting medium • Waves in a uniform conducting guide – Simple example TE01 mode – Propagation constant, cut-off frequency – Group velocity, phase velocity – Illustrations 2
- 3. Reading • J.D. Jackson: Classical Electrodynamics • H.D. Young and R.A. Freedman: University Physics (with Modern Physics) • P.C. Clemmow: Electromagnetic Theory • Feynmann Lectures on Physics • W.K.H. Panofsky and M.N. Phillips: Classical Electricity and Magnetism • G.L. Pollack and D.R. Stump: Electromagnetism 3
- 4. Basic Equations from Vector Calculus y F x F x F z F z F y F F z F y F x F F F F F F 1 2 3 1 2 3 3 2 1 3 2 1 , , : curl : divergence , , , vector a For z φ y φ x φ φ x,y,z,t φ , , : gradient , function scalar a For 4 Gradient is normal to surfaces =constant
- 5. Basic Vector Calculus F F F F G F F G G F 2 ) ( ) ( 0 , 0 ) ( 5 S C r d F S d F dS n S d Oriented boundary C n Stokes’ Theorem Divergence or Gauss’ Theorem S V S d F dV F Closed surface S, volume V, outward pointing normal
- 6. What is Electromagnetism? • 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 Equations Relate Electric and Magnetic fields generated by charge and current distributions. 1 , , In vacuum 2 0 0 0 0 c H B E D t D j H t B E B D 0 E = electric field D = electric displacement H = magnetic field B = magnetic flux density = charge density j = current density 0 (permeability of free space) = 4 10-7 0 (permittivity of free space) = 8.854 10-12 c (speed of light) = 2.99792458 108 m/s
- 8. Maxwell’s 1st Equation 0 0 0 1 Q dV S d E dV E E V S V 0 2 0 3 0 4 4 q r dS q S d E r r q E sphere sphere 0 E 8 Equivalent to Gauss’ Flux Theorem: 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 Area integral gives a measure of the net charge enclosed; divergence of the electric field gives the density of the sources.
- 9. Gauss’ law for magnetism: 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. Maxwell’s 2nd Equation 0 B 0 0 S d B B Gauss’ law for magnetism is then a statement that There are no magnetic monopoles
- 10. Equivalent to Faraday’s Law of Induction: (for a fixed circuit C) The electromotive force round a circuit is proportional to the rate of change of flux of magnetic field, through the circuit. Maxwell’s 3rd Equation t B E dt d S d B dt d l d E S d t B S d E C S S S l d E S d B N S Faraday’s Law is the basis for electric generators. It also forms the basis for inductors and transformers.
- 11. Maxwell’s 4th Equation j B 0 t E c j B 2 0 1 I S d j S d B l d B C S S 0 0 Originates from Ampère’s (Circuital) Law : Satisfied by the field for a steady line current (Biot-Savart Law, 1820): r I B r r l d I B 2 4 0 3 0 current line straight a For Ampère Biot
- 12. Need for Displacement Current • Faraday: vary B-field, generate E-field • Maxwell: varying E-field should then produce a B-field, but not covered by Ampère’s Law. 12 Surface 1 Surface 2 Closed loop Current I Apply Ampère to surface 1 (flat disk): line integral of B = 0I Applied to surface 2, line integral is zero since no current penetrates the deformed surface. In capacitor, , so Displacement current density is t E jd 0 dt dE A dt dQ I 0 A ε Q E 0 t E j j j B d 0 0 0 0
- 13. Consistency with Charge Conservation 0 t j dV t dV j dV dt d S d j Charge conservation: Total current flowing out of a region equals the rate of decrease of charge within the volume. 13 From Maxwell’s equations: Take divergence of (modified) Ampère’s equation t j t j E t c j B 0 0 1 0 0 0 0 2 0 Charge conservation is implicit in Maxwell’s Equations
- 14. Maxwell’s Equations in Vacuum 2 0 0 0 0 1 , , c H B E D In vacuum Source-free equations: Source equations 14 Equivalent integral forms (useful for simple geometries) 0 0 t B E B j t E c B E 0 2 0 1 S d E dt d c S d j l d B dt d S d B dt d l d E S d B dV S d E 2 0 0 1 0 1
- 15. Example: Calculate E from B 0 0 0 0 sin r r r r t B Bz dS B dt d l d E t r B E t B r t B r dt d rE r r cos 2 1 cos sin 2 0 0 2 0 2 0 t r B r E t B r t B r dt d rE r r cos 2 cos sin 2 0 2 0 0 2 0 0 2 0 0 Also from t B E dt E c j B 2 0 1 then gives current density necessary to sustain the fields r z
- 16. Lorentz Force Law • Supplement to Maxwell’s equations, gives force on a charged particle moving in an electromagnetic field: • For continuous distributions, have a force density • Relativistic equation of motion – 4-vector form: – 3-vector component: B v E q f dt p d dt dE c f c f v d dP F , 1 , 16 B j E fd B v E q f v m dt d 0
- 17. Motion of charged particles in constant magnetic fields B v q v m dt d B v E q f v m dt d 0 0 17 1. Dot product with v: constant is constant is 0 So 1 But 0 2 2 2 0 v dt d dt d v dt d v c v γ B v v m q v dt d v No acceleration with a magnetic field constant , 0 0 // 0 v v B dt d B v B m q v dt d B 2. Dot product with B:
- 18. Motion in constant magnetic field 0 0 0 2 0 frequency angular at radius with motion circular m m m qB v ω qB v m ρ B v m q v B v m q dt v d Constant magnetic field gives uniform spiral about B with constant energy. rigidity Magnetic 0 q p q v m B
- 19. Motion in constant Electric Field Solution of E m q v dt d 0 c m qE t m qE 0 2 0 for 2 1 19 Constant E-field gives uniform acceleration in straight line E q v m dt d B v E q f v m dt d 0 0 2 0 2 2 0 1 1 t m qE c v t m qE v is 1 1 2 0 2 0 c m qEt qE c m x v dt dx qEx Energy gain is
- 20. Potentials • Magnetic vector potential: • Electric scalar potential: • Lorentz Gauge: A B A B that such 0 A A f(t) , 20 0 1 2 A t c t A E t A E t A E t A A t t B E so , with 0 Use freedom to set
- 21. Electromagnetic 4-Vectors 21 Α , 1 , 1 0 1 4 2 A c t c A t c Lorentz Gauge 4-gradient 4 4-potential A Current 4-vector 0 0 0 where ) , ( ) , ( j c v c V J v j Continuity equation 0 , , 1 4 j t j c t c J Charge-current transformations 2 , c j v v j j x x x
- 22. Example: Electromagnetic Field of a Single Particle • Charged particle moving along x-axis of Frame F • P has • In F, fields are only electrostatic (B=0), given by t c vx t t t v b r x b vt x p p P 2 2 2 2 ' , ' ' so ), , 0 , ' ( ' 3 3 3 ' ' , 0 ' , ' ' ' ' ' ' r qb E E r qvt E x r q E z y x P Origins coincide at t=t=0 Observer P z b charge q x Frame F v Frame F’ z’ x’ t v x t v x x P P P so ) ( 0
- 23. Electromagnetic Energy • Rate of doing work on unit volume of a system is • Substitute for j from Maxwell’s equations and re-arrange into the form H B D E t S H E S t D E t B H S t D E E H H E E t D H E j 2 1 where 23 E j E v B j E v f v d Poynting vector
- 24. 24 H E D E H B t E j 2 1 S d H E dV H B D E dt d dt dW 2 1 electric + magnetic energy densities of the fields Poynting vector gives flux of e/m energy across boundaries Integrated over a volume, have energy conservation law: rate of doing work on system equals rate of increase of stored electromagnetic energy+ rate of energy flow across boundary.
- 25. Review of Waves • 1D wave equation is with general solution • Simple plane wave: 2 2 2 2 2 1 t u v x u ) ( ) ( ) , ( x vt g x vt f t x u x k t x k t sin : 3D sin : 1D k 2 Wavelength is Frequency is 2
- 26. Superposition of plane waves. While shape is relatively undistorted, pulse travels with the group velocity Phase and group velocities k t x v x k t p 0 dk e k A kx t k i ) ( ) ( dk d vg Plane wave has constant phase at peaks x k t sin 2 x k t
- 27. Wave packet structure • Phase velocities of individual plane waves making up the wave packet are different, • The wave packet will then disperse with time 27
- 28. Electromagnetic waves • Maxwell’s equations predict the existence of electromagnetic waves, later discovered by Hertz. • No charges, no currents: 0 0 B D t B E t D H 2 2 2 2 2 2 2 2 2 : equation wave 3D t E z E y E x E E 2 2 2 2 t E t D B t t B E E E E E 2 2
- 29. Nature of Electromagnetic Waves • A general plane wave with angular frequency travelling in the direction of the wave vector k has the form • Phase = 2 number of waves and so is a Lorentz invariant. • Apply Maxwell’s equations )] ( exp[ )] ( exp[ 0 0 x k t i B B x k t i E E i t k i B E k B E B k E k B E 0 0 Waves are transverse to the direction of propagation, and and are mutually perpendicular B E , k x k t
- 31. Plane Electromagnetic Waves E c B k t E c B 2 2 1 2 Frequency k 2 Wavelength 2 that deduce with Combined kc k B E B E k c k is in vacuum wave of speed Reminder: The fact that is an invariant tells us that is a Lorentz 4-vector, the 4-Frequency vector. Deduce frequency transforms as x k t k c , v c v c k v
- 32. Waves in a Conducting Medium • (Ohm’s Law) For a medium of conductivity , • Modified Maxwell: • Put E j D t E E t E j H E i E H k i conduction current displacement current )] ( exp[ )] ( exp[ 0 0 x k t i B B x k t i E E 4 0 8 - 12 0 7 10 57 . 2 1 . 2 , 10 3 : Teflon 10 , 10 8 . 5 : Copper D D Dissipation factor
- 33. Attenuation in a Good Conductor 0 since with Combine 2 2 E k i k E i E k k E k E i H k E k k H E k t B E E i E H k i For a good conductor D >> 1, i k i k 1 2 , 2 depth - skin the is 2 where 1 1 , exp exp is form Wave i k x x t i copper.mov water.mov
- 34. Charge Density in a Conducting Material • Inside a conductor (Ohm’s law) • Continuity equation is • Solution is E j t E t j t 0 0 t e 0 So charge density decays exponentially with time. For a very good conductor, charges flow instantly to the surface to form a surface charge density and (for time varying fields) a surface current. Inside a perfect conductor () E=H=0
- 35. Maxwell’s Equations in a Uniform Perfectly Conducting Guide 0 2 2 2 2 H E E H i E E E E i t D H H i t B E z y x Hollow metallic cylinder with perfectly conducting boundary surfaces Maxwell’s equations with time dependence exp(it) are: Assume ) ( ) ( ) , ( ) , , , ( ) , ( ) , , , ( z t i z t i e y x H t z y x H e y x E t z y x E Then 0 ) ( 2 2 2 H E t is the propagation constant Can solve for the fields completely in terms of Ez and Hz
- 36. Special cases • Transverse magnetic (TM modes): – Hz=0 everywhere, Ez=0 on cylindrical boundary • Transverse electric (TE modes): – Ez=0 everywhere, on cylindrical boundary • Transverse electromagnetic (TEM modes): – Ez=Hz=0 everywhere – requires 36 0 n Hz i or 0 2 2
- 37. Cut-off frequency, c c gives real solution for , so attenuation only. No wave propagates: cut- off modes. c gives purely imaginary solution for , and a wave propagates without attenuation. For a given frequency only a finite number of modes can propagate. 37 a n e a x n A E a n c z t i c , sin , 1 2 a n a n c For given frequency, convenient to choose a s.t. only n=1 mode occurs. 2 1 2 2 2 1 2 2 1 , c c k ik
- 38. Propagated Electromagnetic Fields From kz t a x n a n A H H kz t a x n Ak H E i H A t B E z y x sin cos 0 cos sin real, is assuming , 38 z x
- 39. Phase and group velocities in the simple wave guide 39 2 1 2 2 c k Wave number: wavelength space free the , 2 2 k Wavelength: velocity space - free than larger , 1 k vp Phase velocity: velocity space - free than smaller 1 2 2 2 k dk d v k g c Group velocity:
- 40. Calculation of Wave Properties • If a=3 cm, cut-off frequency of lowest order mode is • At 7 GHz, only the n=1 mode propagates and GHz 5 03 . 0 2 10 3 2 1 2 8 a f c c c k v c k v k k g p c 1 8 1 8 1 8 9 2 1 2 2 2 1 2 2 ms 10 1 . 2 ms 10 3 . 4 cm 6 2 m 103 10 3 / 10 5 7 2 40 a n c
- 41. Flow of EM energy along the simple guide kz t a x n A a n H H E k H kz t a x n A E E E z y y x y z x sin cos , 0 , cos sin , 0 2 2 2 2 2 2 2 2 0 2 2 0 2 since 8 1 4 1 energy Magnetic 8 1 4 1 energy Electric a n k W k a n a A dx H W a A dx E W e a m a e 41 Fields (c) are: Time-averaged energy: Total e/m energy density a A W 2 4 1
- 42. Poynting Vector 42 Poynting vector is x y z y H E H E H E S , 0 , Time-averaged: a x n kA S 2 2 sin 1 , 0 , 0 2 1 Integrate over x: 2 4 1 akA Sz So energy is transported at a rate: g m e z v k W W S Electromagnetic energy is transported down the waveguide with the group velocity Total e/m energy density a A W 2 4 1