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

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2 Contents Review of Maxwells 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 TE 01 mode –Propagation constant, cut-off frequency –Group velocity, phase velocity –Illustrations

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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

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4 Basic Equations from Vector Calculus Gradient is normal to surfaces =constant

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5 Basic Vector Calculus Oriented boundary C Stokes Theorem Divergence or Gauss Theorem Closed surface S, volume V, outward pointing normal

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What is Electromagnetism? The study of Maxwells 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, Maxwells equations are perfectly consistent with the transformations of special relativity.

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Maxwells Equations Relate Electric and Magnetic fields generated by charge and current distributions. 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 10 8 m/s

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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 Maxwells 1 st Equation Area integral gives a measure of the net charge enclosed; divergence of the electric field gives the density of the sources.

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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. Maxwells 2 nd Equation There are no magnetic monopoles Gauss law for magnetism is then a statement that There are no magnetic monopoles

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Equivalent to Faradays 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. Maxwells 3 rd Equation N S Faradays Law is the basis for electric generators. It also forms the basis for inductors and transformers.

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Maxwells 4 th Equation Originates from Ampères (Circuital) Law : Satisfied by the field for a steady line current (Biot-Savart Law, 1820): Ampère Biot

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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 Ampères Law. Surface 1 Surface 2 Closed loop Current I Apply Ampère to surface 1 (flat disk): line integral of B = 0 I Applied to surface 2, line integral is zero since no current penetrates the deformed surface. In capacitor,, so Displacement current density is

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13 Consistency with Charge Conservation Charge conservation: Total current flowing out of a region equals the rate of decrease of charge within the volume. From Maxwells equations: Take divergence of (modified) Ampères equation Charge conservation is implicit in Maxwells Equations

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14 Maxwells Equations in Vacuum In vacuum Source-free equations: Source equations Equivalent integral forms (useful for simple geometries)

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Example: Calculate E from B Also from then gives current density necessary to sustain the fields r z

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16 Lorentz Force Law Supplement to Maxwells 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:

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17 Motion of charged particles in constant magnetic fields 1.Dot product with v: No acceleration with a magnetic field 2.Dot product with B:

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Motion in constant magnetic field Constant magnetic field gives uniform spiral about B with constant energy.

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19 Motion in constant Electric Field Solution of Constant E-field gives uniform acceleration in straight line is Energy gain is

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According to observer O in frame F, particle has velocity v, fields are E and B and Lorentz force is 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 Relativistic Transformations of E and B Rough idea Exact :

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21 Potentials Magnetic vector potential: Electric scalar potential: Lorentz Gauge: Use freedom to set

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22 Electromagnetic 4-Vectors Lorentz Gauge 4-gradient 4 4-potential A Current 4-vector Continuity equation Charge-current transformations

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23 Relativistic Transformations 4-potential vector: Lorentz transformation Fields:

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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 Origins coincide at t=t =0 Observer P z b charge q x Frame F v z x

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Transform to laboratory frame F: At non-relativistic energies, 1, restoring the Biot- Savart law:

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26 Electromagnetic Energy Rate of doing work on unit volume of a system is Substitute for j from Maxwells equations and re-arrange into the form Poynting vector

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27 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.

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Review of Waves 1D wave equation is with general solution Simple plane wave: Wavelength is Frequency is

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Superposition of plane waves. While shape is relatively undistorted, pulse travels with the group velocitygroup velocity Phase and group velocities Plane wave has constant phase at peaks

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

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Electromagnetic waves Maxwells equations predict the existence of electromagnetic waves, later discovered by Hertz. No charges, no currents:

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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 Maxwells equations Waves are transverse to the direction of propagation, and and are mutually perpendicular

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33 Plane Electromagnetic Wave

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Plane Electromagnetic Waves Reminder: The fact that is an invariant tells us that is a Lorentz 4-vector, the 4-Frequency vector. Deduce frequency transforms as

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Waves in a Conducting Medium (Ohms Law) For a medium of conductivity, Modified Maxwell: Put conduction current displacement current Dissipation factor

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Attenuation in a Good Conductor For a good conductor D >> 1, copper.movcopper.mov water.movwater.mov

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Charge Density in a Conducting Material Inside a conductor (Ohms law) Continuity equation is Solution is 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

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Maxwells Equations in a Uniform Perfectly Conducting Guide z y x Hollow metallic cylinder with perfectly conducting boundary surfaces Maxwells equations with time dependence exp(i t) are: Assume Then is the propagation constant Can solve for the fields completely in terms of E z and H z

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39 Special cases Transverse magnetic (TM modes): – H z =0 everywhere, E z =0 on cylindrical boundary Transverse electric (TE modes): – E z =0 everywhere, on cylindrical boundary Transverse electromagnetic (TEM modes): – E z =H z =0 everywhere –requires

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40 A simple model: Parallel Plate Waveguide Transport between two infinite conducting plates (TE 01 mode): To satisfy boundary conditions, E=0 on x=0 and x=a, so Propagation constant is z x y x=0 x=a

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41 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. For given frequency, convenient to choose a s.t. only n=1 mode occurs.

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42 Propagated Electromagnetic Fields From z x

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43 Phase and group velocities in the simple wave guide Wave number: Wavelength: Phase velocity: Group velocity:

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44 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

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45 Waveguide animations TE1 mode above cut-off ppwg_1-1.movppwg_1-1.mov TE1 mode, smaller ppwg_1-2.mov ppwg_1-2.mov TE1 mode at cut-off ppwg_1-3.movppwg_1-3.mov TE1 mode below cut-off ppwg_1-4.movppwg_1-4.mov TE1 mode, variable ppwg_1_vf.mov ppwg_1_vf.mov TE2 mode above cut-offppwg_2-1.movppwg_2-1.mov TE2 mode, smaller ppwg_2-2.movppwg_2-2.mov TE2 mode at cut-offppwg_2-3.movppwg_2-3.mov TE2 mode below cut-offppwg_2-4.movppwg_2-4.mov

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46 Flow of EM energy along the simple guide Fields ( c ) are: Time-averaged energy: Total e/m energy density

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47 Poynting Vector Poynting vector is Time-averaged: Integrate over x: So energy is transported at a rate: Electromagnetic energy is transported down the waveguide with the group velocity Total e/m energy density

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