Electromagnetism

Contents*Review of Maxwells equations and Lorentz Force LawMotion 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

Reading*J.D. Jackson: Classical ElectrodynamicsH.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 Magnetism G.L. Pollack and D.R. Stump: Electromagnetism

*Basic Equations from Vector CalculusGradient is normal to surfaces =constant

*Basic Vector Calculus

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, Ampre, Biot, Savart and others.

Remarkably, Maxwells equations are perfectly consistent with the transformations of special relativity.

Maxwells 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 10-7 0 (permittivity of free space) = 8.854 10-12 c (speed of light) = 2.99792458 108 m/s

Maxwells 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 field

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

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 2nd Equation Gauss law for magnetism is then a statement thatThere are no magnetic monopoles

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 3rd Equation Faradays Law is the basis for electric generators. It also forms the basis for inductors and transformers.

Maxwells 4th EquationOriginates from Ampres (Circuital) Law :

Satisfied by the field for a steady line current (Biot-Savart Law, 1820):

Need for Displacement CurrentFaraday: vary B-field, generate E-fieldMaxwell: varying E-field should then produce a B-field, but not covered by Ampres Law.*

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) Ampres equation

Charge conservation is implicit in Maxwells Equations

Maxwells Equations in Vacuum*In vacuum

Source-free equations:

Source equationsEquivalent integral forms (useful for simple geometries)

Example: Calculate E from B

Lorentz Force LawSupplement 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:*

*Motion of charged particles in constant magnetic fieldsNo acceleration with a magnetic field

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

*Motion in constant Electric FieldSolution of Constant E-field gives uniform acceleration in straight lineis

Relativistic Transformations of E and BAccording 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

SuggestsRough idea

PotentialsMagnetic vector potential:

Electric scalar potential:

Lorentz Gauge:*Use freedom to set

Electromagnetic 4-Vectors*

*Relativistic Transformations4-potential vector:

Lorentz transformation

Fields:

Example: Electromagnetic Field of a Single ParticleCharged particle moving along x-axis of Frame F

P has

In F, fields are only electrostatic (B=0), given byOrigins coincide at t=t=0Observer P

Transform to laboratory frame F:

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

*electric + magnetic energy densities of the fieldsPoynting vector gives flux of e/m energy across boundariesIntegrated 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.

Review of Waves1D wave equation is with general solution

Simple plane wave:

Superposition of plane waves. While shape is relatively undistorted, pulse travels with the group velocityPhase and group velocities

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

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

Nature of Electromagnetic WavesA 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

Plane Electromagnetic Wave*

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

Waves in a Conducting Medium(Ohms Law) For a medium of conductivity , Modified Maxwell:

Put

Attenuation in a Good Conductorcopper.mov water.mov

Charge Density in a Conducting MaterialInside a conductor (Ohms law)Continuity equation is

Solution 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

Maxwells Equations in a Uniform Perfectly Conducting GuideHollow metallic cylinder with perfectly conducting boundary surfacesMaxwells equations with time dependence exp(iwt) are:g is the propagation constantCan solve for the fields completely in terms of Ez and Hz

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 everywhererequires

*A simple model: Parallel Plate WaveguideTransport between two infinite conducting plates (TE01 mode):

Cut-off frequency, wcwwc 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.

*Propagated Electromagnetic FieldsFrom

*Phase and group velocities in the simple wave guide

*Calculation of Wave PropertiesIf a=3cm, cut-off frequency of lowest order mode is

At 7GHz, only the n=1 mode propagates and

Waveguide animations*TE1 mode above cut-off ppwg_1-1.movTE1 mode, smaller ppwg_1-2.movTE1 mode at cut-off ppwg_1-3.movTE1 mode below cut-off ppwg_1-4.movTE1 mode, variable ppwg_1_vf.movTE2 mode above cut-offppwg_2-1.movTE2 mode, smaller ppwg_2-2.movTE2 mode at cut-offppwg_2-3.movTE2 mode below cut-offppwg_2-4.mov

*Flow of EM energy along the simple guideFields (w>wc) are:Time-averaged energy:

Poynting Vector*Electromagnetic energy is transported down the waveguide with the group velocity

*

*******1831-1879.1861 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 worlds 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 worlds first colour photograph. Maxwells 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. ***Michael Faraday (1791-1867). 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 1831. 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.*Andre Marie Ampere (1775-1836), 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 1802. Published treatises on Mathematical Theory of Games and Calculus of Variations (1803). Appointed tutor at Ecole Polytechnique in Paris in 1804. Professor in 1809, chair at Universite de France in 1826. Disastrous marriage 1806, separated 1807. 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 Orsteds experimental work. Discovered electrodynamical forces between linear wires. Similar work also done by Biot and Savart, and by Poisson. Amperes 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. Amperes son achieved fame as a historian, his daughter married one of Napoleons lieutenants, who became an alcoholic and the marriage disintegrated acrimoniously.

Biot: 1774-1862, 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). 1809 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.***********************************