Presentation on theme: "Electromagnetism Christopher R Prior ASTeC Intense Beams Group"— Presentation transcript:
1 Electromagnetism Christopher R Prior ASTeC Intense Beams Group Fellow and Tutor in MathematicsTrinity College, OxfordASTeC Intense Beams GroupRutherford Appleton Laboratory
2 Contents Review of Maxwell’s equations and Lorentz Force Law Motion of a charged particle under constant Electromagnetic fieldsRelativistic transformations of fieldsElectromagnetic energy conservationElectromagnetic wavesWaves in vacuoWaves in conducting mediumWaves in a uniform conducting guideSimple example TE01 modePropagation constant, cut-off frequencyGroup velocity, phase velocityIllustrations
3 Reading J.D. Jackson: Classical Electrodynamics H.D. Young and R.A. Freedman: University Physics (with Modern Physics)P.C. Clemmow: Electromagnetic TheoryFeynmann Lectures on PhysicsW.K.H. Panofsky and M.N. Phillips: Classical Electricity and MagnetismG.L. Pollack and D.R. Stump: Electromagnetism
4 Basic Equations from Vector Calculus Gradient is normal to surfaces =constant
5 Divergence or Gauss’ Theorem Basic Vector CalculusOriented boundary CStokes’ TheoremDivergence or Gauss’ TheoremClosed 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 EquationsRelate Electric and Magnetic fields generated by charge and current distributions.E = electric fieldD = electric displacementH = magnetic fieldB = magnetic flux density= charge densityj = current density0 (permeability of free space) = 4 10-70 (permittivity of free space) =c (speed of light) = m/s1861 Maxwell had the great idea that unified electricity and magnetism. Made astounding prediction that fleeting electric currents could exist not only in conductors but in all materials, even empty space. Fitted everything together into a single theory. Theory predicted that every time a magnet jiggled or an electric current changed, a wave of energy spread out. He found that waves travelled with speed of light. At a stroke unified electricity, magnetism and light. Astonished everyone, proof took 25 years, when Heinrich Hertz produced waves from a spark-gap source and detected them. Now one of the central pillars of our understanding of the universe, and opened the way to 20th century relativity and quantum theory.Maxwell would have been among the world’s greatest scientists even without his work in electricity and magnetism. His influence is everywhere. He introduced statistical methods into physics; he demonstrated the principle by which we see colours and took the world’s first colour photograph. Maxwell’s demon – molecule-sized creature that could make heat flow from a hot to a cold gas – was the first effective scientific thought experiment. He stimulated the creation of information theory; he wrote a paper on automated control systems that became the foundation of modern control theory and cybernetics. He showed how to use polarised light to reveal strain patterns in a structure; he was the first to suggest a centrifuge to separate gases.
8 Maxwell’s 1st Equation 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 fieldArea integral gives a measure of the net charge enclosed; divergence of the electric field gives the density of the sources.
9 Maxwell’s 2nd Equation 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.Gauss’ law for magnetism is then a statement that There are no magnetic monopoles
10 Maxwell’s 3rd Equation 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.NSMichael Faraday ( ). Apprenticed to a London bookbinder. Became fascinated in science by reading books in the shop. After writing to Davy, was given a job as a lab assistant at the Royal Institution. Became a skilled Chemist. In 1823 discovered that chlorine could be liquefied, also discovered benzene. Discovered e/m induction in His work laid the foundations of all subsequent e/m, leading to devices such as the transformer and electric motor. Was one of the greatest scientific lecturers of the age. Often said to be the greatest experimentalist in the history of science.Faraday’s Law is the basis for electric generators. It also forms the basis for inductors and transformers.
11 Maxwell’s 4th Equation Originates from Ampère’s (Circuital) Law : Satisfied by the field for a steady line current (Biot-Savart Law, 1820):AmpèreBiotAndre Marie Ampere ( ), son of a wealthy family from Lyon in France. Did not attend formal school, but was taught at home, mainly by his father. Claimed to have mastered all known mathematics by the age of 12. Submitted first paper at age of 13 (as he was unaware of calculus, this was not thought worthy of publication). Family was badly hit by French revolution (sister died, father guillotined). Became Prof. of Physics and Chemistry at Bourg Ecole Centrale in Published treatises on Mathematical Theory of Games and Calculus of Variations (1803). Appointed tutor at Ecole Polytechnique in Paris in Professor in 1809, chair at Universite de France in Disastrous marriage 1806, separated Worked on partial differential equations, chemistry (fluorine, classification of elements); theory of light (refraction, advocate of wave theory) and a combined theory of electricity and magnetism, produced very quickly after hearing of the Danish physicist Orsted’s experimental work. Discovered electrodynamical forces between linear wires. Similar work also done by Biot and Savart, and by Poisson. Ampere’s most important publication, Memoir on the Mathematical Theory of Electrodynamic Phenomena, was published in 1826, and his theories became fundamental for 19th century developments in electricity and magnetism. Also discovered induction 9 years before Faraday but let the latter have full credit. Ampere’s son achieved fame as a historian, his daughter married one of Napoleon’s lieutenants, who became an alcoholic and the marriage disintegrated acrimoniously.Biot: , Paris.Education at Louis-Le-Grand, followed by the army. Took part in insurrection against government, captured but released after pleadings by Monge. Professor of Mathematics at Beauvais, then at College de France (thanks to Laplace) Professor of Astronomy in Spain. Made advances in astronomy, elasticity, electricity and magnetism, heat and optics; mathematical work in geometry. Together with Savart, discovered that intensity of magnetic field set up by a current in a wire varies inversely with the distance from the wire. Awarded the Rumford Medal of the Royal Society for his work on the polarisation of light passing through chemical solutions.
12 Need for Displacement Current Faraday: vary B-field, generate E-fieldMaxwell: varying E-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 = 0IApplied to surface 2, line integral is zero since no current penetrates the deformed surface.In capacitor, , soDisplacement current density isSurface 1Surface 2Closed loopCurrent I
13 Consistency with Charge Conservation Total current flowing out of a region equals the rate of decrease of charge within the volume.From Maxwell’s equations: Take divergence of (modified) Ampère’s equationCharge conservation is implicit in Maxwell’s Equations
14 Maxwell’s Equations in Vacuum Source-free equations:Source equationsEquivalent integral forms (useful for simple geometries)
15 Example: Calculate E from B zAlso fromthen gives current density necessary to sustain the fields
16 Lorentz Force LawSupplement to Maxwell’s equations, gives force on a charged particle moving in an electromagnetic field:For continuous distributions, have a force densityRelativistic equation of motion4-vector form:3-vector component:
17 Motion of charged particles in constant magnetic fields Dot product with v:No acceleration with a magnetic fieldDot product with B:
18 Motion in constant magnetic field Constant magnetic field gives uniform spiral about B with constant energy.
19 Motion in constant Electric Field Solution ofisEnergy gain isConstant E-field gives uniform acceleration in straight line
20 Relativistic Transformations of E and B According to observer O in frame F, particle has velocity v, fields are E and B and Lorentz force isIn Frame F, particle is at rest and force isAssume measurements give same charge and force, soPoint charge q at rest in F:See a current in F, giving a fieldSuggestsExact:Rough idea
21 Potentials Magnetic vector potential: Electric scalar potential: Lorentz Gauge:Use freedom to set
24 Example: Electromagnetic Field of a Single Particle Charged particle moving along x-axis of Frame FP hasIn F, fields are only electrostatic (B=0), given byzbcharge qxFrame FvFrame F’z’x’Observer POrigins coincide at t=t=0
25 Transform to laboratory frame F: At non-relativistic energies, ≈ 1, restoring the Biot-Savart law:
26 Electromagnetic Energy Rate of doing work on unit volume of a system isSubstitute for j from Maxwell’s equations and re-arrange into the formPoynting vector
27 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.electric + magnetic energy densities of the fieldsPoynting vector gives flux of e/m energy across boundaries
28 Review of Waves 1D wave equation is with general solution Simple plane wave:Wavelength isFrequency is
29 Phase and group velocities Superposition of plane waves. While shape is relatively undistorted, pulse travels with the group velocityPlane wave has constant phase at peaks
30 Wave packet structurePhase velocities of individual plane waves making up the wave packet are different,The wave packet will then disperse with time
31 Electromagnetic waves Maxwell’s equations predict the existence of electromagnetic waves, later discovered by Hertz.No charges, no currents:
32 Nature of Electromagnetic Waves A general plane wave with angular frequency travelling in the direction of the wave vector k has the formPhase = 2 number of waves and so is a Lorentz invariant.Apply Maxwell’s equationsWaves are transverse to the direction of propagation, and and are mutually perpendicular
34 Plane Electromagnetic Waves Reminder: The fact that is an invariant tells us thatis a Lorentz 4-vector, the 4-Frequency vector. Deduce frequency transforms as
35 Waves in a Conducting Medium (Ohm’s Law) For a medium of conductivity ,Modified Maxwell: Putconduction currentdisplacement currentDissipation factor
36 Attenuation in a Good Conductor For a good conductor D >> 1,copper.mov water.mov
37 Charge Density in a Conducting Material Inside a conductor (Ohm’s law)Continuity equation isSolution isSo 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
38 Maxwell’s Equations in a Uniform Perfectly Conducting Guide Hollow metallic cylinder with perfectly conducting boundary surfaceszyxMaxwell’s equations with time dependence exp(iwt) are:AssumeTheng is the propagation constantCan solve for the fields completely in terms of Ez and Hz
39 Special cases Transverse magnetic (TM modes): Hz=0 everywhere, Ez=0 on cylindrical boundaryTransverse electric (TE modes):Ez=0 everywhere, on cylindrical boundaryTransverse electromagnetic (TEM modes):Ez=Hz=0 everywhererequires
40 A simple model: “Parallel Plate Waveguide” Transport between two infinite conducting plates (TE01 mode):zxyx=0x=aTo satisfy boundary conditions, E=0 on x=0 and x=a, soPropagation constant is
41 For given frequency, convenient to choose a s.t. only n=1 mode occurs. Cut-off frequency, wcw<wc gives real solution for g, so attenuation only. No wave propagates: cut-off modes.w>wc gives purely imaginary solution for g, and a wave propagates without attenuation.For a given frequency w only a finite number of modes can propagate.For given frequency, convenient to choose a s.t. only n=1 mode occurs.
46 Flow of EM energy along the simple guide Fields (w>wc) are:Total e/m energy densityTime-averaged energy:
47 Poynting Vector Poynting vector is Time-averaged: Integrate over x: Total e/m energy densityIntegrate over x:So energy is transported at a rate:Electromagnetic energy is transported down the waveguide with the group velocity