PHY 4966 EM THEORY

General Information This course is aimed at MSci students in the the Physics, Theoretical Physics and Astrophysics pathways.

Aims and Objectives

Aims:

The course is designed to introduce the essential elements of classical electromagnetism, covering Maxwell's equations in vacuo and matter, electromagnetic waves, radiation and scattering, and the Lorentz covariant formulation of the theory.

Objectives:

A student who has satisfactorily completed the course should be able:

A student who has satisfactorily completed the course should be able:

Syllabus & Prerequisites
The syllabus will be as follows. The approximate time to be spent on each set of topics is also indicated:

Revision of laws of electromagnetism in vacuo, displacement current, Maxwell's equations in vacuo, charge and current density sources, energy theorems, fluxes of energy and momentum. [2 hours]
Polarization and magnetization,D and H fields, linear media, boundary conditions on the fields in media, Maxwell stress tensor, concept of macroscopic fields as space averages of molecular fields, Lorentz local field argument, the Clausius- Mossotti relation. [3 hours]
Maxwell's equations in media, Homogeneous wave equation in vacuo and in media, concept of frequency dependent dielectric function, properties of its real and imaginary parts, causality, Kramers-Kronig relation. [3 hours]
Scalar and vector potentials, gauge transformations, inhomogeneous wave equation, the retarded solution to the wave equation, radiation from a Hertzian dipole with discussion of near and far fields, formula for power radiated, qualitative discussion of magnetic dipole and electric quadrupole radiation. [4 hours]
Scattering of a plane wave by a single slowly moving charged particle, total and differential scattering cross-sections, optical theorem, scattering from a medium with space-varying dielectric constant, scattering from an assemblage of polarizable particles, Rayleigh-Smoluchowski-Einstein theory of why the sky is blue - critical opalescence. [5 hours] Lorentz transformations, charge and current density as a 4-vector, the potential 4-vector, tensors and invariants, the relativistic field tensor F, Lorentz transformation properties of current density and potential 4-vectors and of the free vacuum E and B fields, tensor form of Maxwell's equations, covariant formulation of energy and momentum theorems, energy-momentum tensor. [5 hours]
Lienard-Wiechert potentials for a moving charged particle derived from a delta-function source, fields for a uniformly moving charged particle in the non-relativistic and ultra-relativistic limits, radiation from accelerated charges, the cases of velocity and acceleration parallel and perpendicular, Larmor formula for radiated power, bremsstrahlung and synchrotron radiation as examples. [5 hours]
Maxwell theory as a Lagrangian field theory, the free field as an ensemble of oscillators. [3 hours]

Prerequisite Knowledge
Knowledge of electromagnetism which is contained in the current core syllabi of the Physics undergraduate degrees in the London colleges will be assumed. The following additional pre-requisite knowledge in mathematics and physics will also be assumed:

Taylor series.
Div, Grad and Curl, Surface and Volume integrals, Gauss and Stokes theorems.
The complex representation of harmonically varying quantities.
Fourier transforms.
The one-dimensional wave equation.
Matrix multiplication and familiarity with indices.
Contour integration up to Cauchy's theorem (this is used only in the discussion of the Kramers-Kronig relation)
From special relativity the explicit form of the simple Lorentz transformation between frames in relative motion along a single coordinate direction.
It is desirable but not necessary that students have met the Lagrangian formulation of particle mechanics.
We will not assume that students have met the concept of Green functions before.

Sources and textbooks:

  • Classical Electrodynamics, J.D.Jackson, 3rd Edition, Wiley 1998. This is the text for this course, and covers much more material in addition.

  • Classical Electrodynamics , H.C.O'Hanian, Allyn and Bacon 1988. Somewhat simpler and more concise than Jackson. The book is out of print, but held in many libraries.

  • A History of the Theories of Aether and Electricity: The Classical Theories , E. Whittaker, Thomas Nelson and Sons, 2nd Edition 1951. Interesting book on the history of electromagnetism up to 1900.

  • Principles of Optics , M. Born and E.Wolf, 6th Edition, Pergamon 1980. Includes brief discussions of basic properties of the electromagnetic field and polarisation, dispersion and propagation.

  • Electricity and Magnetism , W.N. Cottingham and D.A. Greenwood, Cambridge University Press Wiley 1995. An undergraduate text, but useful for the background to some of the material in this course.

    Lecture Notes:

    The following notes (in PDF format) summarise the course. More detailed discussion of this material will be presented in the lectures, and the course of the lectures may not follow exactly that of the notes. Lecture Notes 7 and 8 will be covered in one week.

    Lecture Notes 1: Historical background, vector calculus, Maxwell's equations, energy and momentum. Magnetic monopoles.

    Lecture Notes 2: Linear media, polarisation and magnetisation, Maxwell's equations in matter, boundary conditions, energy and momentum, the Clausius-Mossotti relation, solved problems.

    Lecture Notes 3: Plane waves, polarisation, dispersion, the Kramers-Kronig relations.

    Lecture Notes 4: Scalar and vector potentials, the inhomogeneous wave equation, the delta function, the Green function.

    Lecture Notes 5: Radiation from a generalised localised source, electric dipole radiation, magnetic dipole radiation and higher order terms, radiation from an antenna.

    Lecture Notes 6: Scattering, scattering from a small scatterer, many scatterers, scattering from the sky, the Born approximation, Rayleigh's explanation for the blue sky, critical opalescence, the optical theorem. Supplementary EMT Notes.

    Lecture Notes 7: Special relativity, four vectors, time dilation and the Lorentz-Fitzgerald contraction, the four-velocity, energy and momentum, covariant and contravariant vectors, tensors. Lecture Notes 8: The charge-current density four-vector, the Lorentz force, the potential four-vector, the field strength tensor, the dual field strength tensor, the energy-momentum tensor.

    Lecture Notes 9: Fields from a static source and a moving charged particle, the Lienard-Wiechert potentials, motion in a circle.

    Lecture Notes 10: The Lagrangian and Hamiltonian for a charged particle and the electromagnetic field, the canonical and symmetric stress tensors, the conservation laws, the field as an ensemble of oscillators.

    Lecture Notes 11: Discussion of two modern fields where topics presented in this course feature:
    (1) The Standard Model. (2) Duality, Gravity and M-Theory.

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