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

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.

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

- To write down Maxwell's equations in vacuo and discuss the individual terms in these equations and their physical significance; understand the definitions and use of the energy and momentum fluxes.
- To know how to generalise the Maxwell equations to the case of linear media, knowing the definitions of the D and H fields, know the boundary conditions applying at matter interfaces, know the main features of the Maxwell stress tensor, and the basic assumptions and result for the Clausius-Mossotti relation.
- Describe the solutions to the homogeneous wave equation in vacuo and in media; describe the physical significance of a frequency dependent dielectric function and its real and imaginary parts; discuss the significance of the Kramers-Kronig relations.
- To construct the electric and magnetic fields from the vector and scalar
potentials and to appreciate the role of gauge invariance; to understand the
derivation of the inhomogeneous wave equation and how the solutions to this
equation can be generated.
To follow the arguments leading to the description and analysis of dipole and multipole radiation, including the concepts of near and far fields and the power formula for electric dipole radiation. To understand the definitions of total and differential scattering cross-sections; to follow arguments leading to the optical theorem; to derive formulas for the scattering of radiation from a small polarisable scatterer, and from a collection of scatterers, and thus to obtain an explanation for the blue colour of the sky; to know what critical opalescence is. - To use 4-vectors and tensors to demonstrate the covariance of Maxwell's equations under the Lorentz transformations of Special Relativity; to understand the covariant formulation of energy and momentum theorems using the energy-momentum tensor.
- To derive the equations of motion for a charged particle moving under the action of given (simple) fields; to obtain the Lienard-Wiechert potentials, and so to deduce the radiation from an accelerated charge; to be aware of applications of the Larmor formula for radiated power;
- To understand the Lagrangian formulation of classical electrodynamics; to know the expression of the free field as an ensemble of oscillators.

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**:

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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