# Electrodynamics

### PHYSICS762

Maxwell's equations, special relativity, covariant formulation of electrodynamics, conservation laws, electrostatics, magnetostatics, boundary conditions, electromagnetic induction, electromagnetic waves, and elementary radiation theory. 3 units.

#### Possible principal texts:

- C. A. Brau,
*Modern Problems in Classical Electrodynamics*, (Oxford Univ. Press, 2004). - F. Melia,
*Electrodynamics*, (U. Chicago Press, Chicago, 2001).

#### Other texts to consider:

- L. D. Landau and E. Lifschitz,
*The Classical Theory of Fields*(vol. 2). - L. D. Landau and E. Lifschitz,
*Electrodynamics of Continuous Media*(vol. 8). - J. D. Jackson,
*Classical Electrodynamics*, (Wiley, New York, 1999). - F. E. Low,
*Classical Field Theory*, (Wiley, New York, 1997).

#### Prerequisites

The prerequisites are:

- at least one semester of an intermediate undergraduate electromagnetism course at the level of
**David J. Griffiths**'s textbook; and - knowledge of mathematical techniques at the level of PHY 301; and
- knowledge of computational techniques, for example
*Mathematica*or*Maple*.

In Duke Physics, there is an undergraduate "Electricity & Magnetism" course (PHY 182), with the synopsis: Electrostatic fields and potentials, boundary value problems, magnetic induction, energy in electromagnetic fields, Maxwell's equations, introduction to electromagnetic radiation.

#### Syllabus

- Special relativity: space-time, Lorentz transformations, time dilation, length contraction, and velocity transformation.
- Cartesian tensors, four-vectors and four-tensors, metric tensor, four-vector calculus.
- Thomas precession and spin.
- Covariant electrodynamics: Four-tensor electromagnetic field, transformation of fields, electric and magnetic fields of a moving charge, Lagrangian for charged particle in a vector potential, Lagrangian density for the electromagnetic field. gauge transformations, stress-energy tensor, conservation laws.
- Electrostatics: multipole expansion, boundary value problems, energetics, solution methods - method of images; electrostatic screening; Green function methods.
- Magnetostatics: magnetic scalar potential, energetics.
- Dielectric and magnetic materials: polarization, boundary conditions, macroscopic form of Maxwell’s equations.
- Faraday’s law, coefficients of inductance, magnetic diffusion.
- Electromagnetic waves: plane waves in vacuum, energy and momentum transport, polarization, plane waves in materials of index n > 1; reflection and refraction at oblique incidence (Fresnel equations), frequency-dependent dielectric function.
- Electromagnetic radiation: Lienard-Wiechert potentials and fields, electric dipole radiation; Larmor formula. multipole radiation; magnetic dipole and electric quadrupole, radiation from relativistic charges.

Special topics might include, e.g., reflection from metallic surfaces, surface plasma waves, Bremstrahlung, or synchrotron radiation.

*Sample* lecture schedule #1

(based on 25 lectures each of duration 75 minutes).

*Lecture 1:*__Review__of undergrad EM, part 1: electrical charge, electric field and potential, current, magnetic field and vector potential, and electromagnetic energy.*Lecture 2:*__Review__of undergrad EM, part 2: Faraday’s law, Maxwell’s equation in vacuum, electromagnetic waves, and Poynting’s Theorem.*Lecture 3:*Special relativity 1: Background review - Space-time, Lorentz transformations, time dilation, length contraction, and velocity transformation.*Lecture 4:*Special relativity 2: Review of Cartesian tensors, four-vectors and four-tensors, metric tensor.*Lecture 5:*Special relativity 3: Four-vector calculus.*Lecture 6:*Special relativity 4: Thomas precession and spin.*Lecture 7:*Covariant electrodynamics 1: Four-tensor electromagnetic field, transformation of fields, electric and magnetic fields of a moving charge.*Lecture 8:*Covariant electrodynamics 2: Lagrangian for charged particle in a vector potential, Lagrangian density for the electromagnetic field.*Lecture 9:*Covariant electrodynamics 3: Gauge transformations, stress-energy tensor, and conservation laws.*Lecture 10:*Electrostatics 1: Multipole expansion of the electrostatic field.*Lecture 11:*Electrostatics 2: Boundary value problems (BVPs) – systems of conductors; coefficients of capacitance; energetics.*Lecture 12:*Electrostatics 3: BVP solution methods - Method of images; electrostatic screening; fields near tips and edges.*Lecture 13:*Electrostatics 4: BVP solution methods: Green function methods.*Lecture 14:*Magnetostatics: Magnetic scalar potential, energetics.*Lecture 15:*Dielectric and magnetic materials (phenomenological treatment): polarization, boundary conditions, macroscopic form of Maxwell’s equations.*Lecture 16:*Faraday’s law, coefficients of inductance, magnetic diffusion.*Lecture 17:*Electromagnetic waves 1: plane waves in vacuum, energy and momentum transport, polarization.*Lecture 18:*Electromagnetic waves 2: plane waves in materials of index n > 1; reflection and refraction at oblique incidence – Fresnel equations.*Lecture 19:*Electromagnetic waves 3: elementary theory of frequency-dependent dielectric function.*Lecture 20:*Electromagnetic radiation 1: Lienard-Wiechert potentials and fields.*Lecture 21:*Electromagnetic radiation 2: Electric dipole radiation; Larmour formula.*Lecture 22:*Electromagnetic radiation 3: Multipole radiation; magnetic dipole and electric quadrupole.*Lecture 23:*Electromagnetic radiation 4: Radiation from relativistic charges.*Lecture 24:*Special topics.*Lecture 25:*Special topics.

Special topics might include, e.g., reflection from metallic surfaces, surface plasma waves, Bremstrahlung, or synchrotron radiation.

*Sample* lecture schedule #2

(based on 25 lectures each of duration 75 minutes).

*Lecture 1:*__Review__of undergrad EM, part 1: electrical charge, electric field and potential, current, magnetic field and vector potential, and electromagnetic energy.*Lecture 2:*__Review__of undergrad EM, part 2: Faraday’s law, Maxwell’s equation in vacuum, electromagnetic waves, and Poynting’s Theorem.*Lecture 3:*Electrostatics 1: Multipole expansion of the electrostatic field.*Lecture 4:*Electrostatics 2: Boundary value problems (BVPs) – systems of conductors; coefficients of capacitance; energetics.*Lecture 5:*Electrostatics 3: BVP solution methods - Method of images; electrostatic screening; fields near tips and edges.*Lecture 6:*Electrostatics 4: BVP solution methods: Green function methods.*Lecture 7:*Magnetostatics: Magnetic scalar potential, energetics.*Lecture 8:*Dielectric and magnetic materials (phenomenological treatment): polarization, boundary conditions, macroscopic form of Maxwell’s equations.*Lecture 9:*Faraday’s law, coefficients of inductance, magnetic diffusion.*Lecture 10:*Electromagnetic waves 1: plane waves in vacuum, energy and momentum transport, polarization.*Lecture 11:*Electromagnetic waves 2: plane waves in materials of index n > 1; reflection and refraction at oblique incidence – Fresnel equations.*Lecture 12:*Electromagnetic waves 3: elementary theory of frequency-dependent dielectric function.*Lecture 13:*Special relativity 1: Background review - Space-time, Lorentz transformations, time dilation, length contraction, and velocity transformation.*Lecture 14:*Special relativity 2: Review of Cartesian tensors, four-vectors and four-tensors, metric tensor.*Lecture 15:*Special relativity 3: Four-vector calculus.*Lecture 16:*Special relativity 4: Thomas precession and spin.*Lecture 17:*Covariant electrodynamics 1: Four-tensor electromagnetic field, transformation of fields, electric and magnetic fields of a moving charge.*Lecture 18:*Covariant electrodynamics 2: Lagrangian for charged particle in a vector potential, Lagrangian density for the electromagnetic field.*Lecture 19:*Covariant electrodynamics 3: Gauge transformations, stress-energy tensor, and conservation laws.*Lecture 20:*Electromagnetic radiation 1: Lienard-Wiechert potentials and fields.*Lecture 21:*Electromagnetic radiation 2: Electric dipole radiation; Larmour formula.*Lecture 22:*Electromagnetic radiation 3: Multipole radiation, magnetic dipole and electric quadrupole.*Lecture 23:*Electromagnetic radiation 4: Radiation from relativistic charges.*Lecture 24:*Special topics*Lecture 25:*Special topics