Topics covered: Electrodynamics-II, Autumn 2011
- Lecture 1 (Aug 3):
- Introduction to the course and course contents
- Steady-state Maxwell equations in vacuum and matter: a reminder
- Lecture 1 notes
- Lecture 2 (Aug 5):
- Uniqueness theorem for a vector whose Div and Grad is given throughout
the space
- Uniqueness theorem for a scalar potential
- Method of images and the relevance of the uniqueness theorems
- Solving Laplace's equation with specific boundary conditions
- Cartesian coordinates: travelling/ oscillating and hyperbolic/decaying
solutions
- Spherical coordinates ...
- Lecture 2 notes
- Lecture 3 (Aug 10):
- Spherical coordinates: spherical harmonics, associated Legendre
polynomials, Legendre polynomials
- Polar coordinates: Bessel functions
- Uniqueness theorem for vector potential
- Energy stored in an electric field
- Lecture 3 notes
- Lecture 4 (Aug 12):
- Energy stored in a magnetic field
- Relaxation of currents and fields inside a conductor
- Electromagnetic waves in a medium: propagating and dispersing solutions
- Plane waves, Longitudinal and transverse components of
electric and magnetic fields
- Poynting vetor
- Lecture 4 notes
- Drop Test (Aug 13)
- Lecture 5 (Aug 17):
- Reflection, refraction, transmission at dielectric boundaries
- Total internal reflection, Brewster's angle,
- Propagation / decay of EM waves inside a conductor
- Boundary conditions at a conducting surface
- Lecture 5 notes
- Lecture 6 (Aug 19):
- Waveguides: rectangular and circular cylindrical, TE and TM modes
- Minimum frequency for propagation, phase velocity, group velocity
- Coaxial cable, TEM waves
- Cavities: standing waves
- Lecture 6 notes
- Mathematica Tutorial by
Rahul Dandekar (Aug 26)
- Lecture 7 (Aug 31):
- Review of waveguides, coaxial cable and cavities
- Transmitted power, conductance
- Potentials [A] and [phi], gauge condition
- Lorentz condition, wave equations for [A] and [phi] in the
presence of sources
- Use of Fourier Transform and Green's function method for
solving the wave equation
- Advanced and retarded solutions for [A] and [phi]
- Lecture 7 notes
- Lecture 8 (Sep 2):
- The fields [E] and [B] from the retarded potentials [A] and [phi]
- EM Radiation: (1/r) dependence for [E] and [B]
- Poynting vector, energy radiated at a given frequency
- Total energy radiated per solid angle
- Lecture 8 notes
- Assignment 1 (Given Saturday Sep 3,
Expected Wednesday Sep 21)
- Lecture 9 (Sep 14):
- Multipole expansion in the limit [ x' < lambda < r ]
- Electric dipole radiation: omega dependence, angular distribution
- Separation of magnetic dipole and electric quadrupole terms
- Electric quadrupole radiation: similarity with gravitational waves
- Lecture 9 notes
- Lecture 10 (Sep 16):
- Faraday's law, caveats, and Lorentz force
- More motivations for special relativity: speed of light,
consistency of Maxwell's equations
- Spacetime transformations from linearity and constancy of speed of light
- Length contraction and time dilation
- Lecture 11 (Sep 21):
- Transformation properties of [E] and [B] fields, given the invariance
of Maxwell's equations under Lorentz transformations
- [E] and [B] fields of a free EM wave satisfy the wave equation
(speed c) in all frames
- Aberration of light
- Doppler shift
- Lecture 12 (Sep 23):
- Velocity transformation / addition
- Definition of momentum through momentum conservation in elastic collision
- Relativistic kinetic energy using work done on a moving body
- Modification of Einstein's argument that suggests [E = mc2]
- Force and acceleration in relativity
- Compton scattering and particle decay: conservation of relativistic
energy and momentum
- Lecture 13 (Sep 28):
- Boosting in an arbitrary direction
- (E/c, p), (c rho, J), (phi/c, A) having the same transformation
properties as (ct, x): contravariant quantities
- A different transformation property of (d/dt, \grad): covariance
- 4-vectors, covariant and contravariant components
- scalar product / Lorents invariants: distance, mass, phase,
D'alembertian, continuity equation, Lorentz gauge
- Metric as the matrix transforming covariant to contravariant,
as the quantity for defining scalar products
- Assignment 2 (Given Thursday Sep 29,
Expected Wednesday Oct 19)
- Lecture 14 (Sep 30):
- Proper disyance ds between two spacetime points: spacelike and timelike
separation, causality
- 4-velocity [u] and 4-acceleration [a]
- [AB -> CD] scattering and [A -> BC] decay: Lorentz invariants and
relations among them
- Symmetric second rank tensors: [Lambda] and [g]
- Antisymmetric second rank tensor: a vector and an axial vector as
subcomponents
- Lecture 15 (Oct 5):
- Electromagnetif field tensor [F] as a gauge invariant quantity
- [F] in terms of electric and magnetic fields
- Fourth rank completely antisymmetric tensor [epsilon]
- Dual tensors with [epsilon]
- Lorentz invariants [F_ik F^ik] and [Ftilde_ik F^ik]
- Maxwell equation without sources: [del Ftilde = 0] as an identity
- Tutorial (Oct 7)
- Midterm Exam (Oct 8)
- Lecture 16 (Oct 12):
- Maxwell's equations in covariant notation
- Line element, area element, 3-volume element and 4-volume element
in covariant notation
- Gauss's theorem and Stokes' theorem in covariant notation
- Maxwell's equations in integral form: covariant notation
- Lecture 17 (Oct 14):
- Principle of least action
- Review of Lagrangian, momentum, Hamiltonian, equations of motion
- Non-relativistic Lagrangian with a potential term
- Relativistic free particle: Lagrangian [- m/gamma ds],
momentum, Hamiltonian, equations of motion in 3-vector and 4-vector notation
- EM interaction term [(e/c) A_i dx^i] added,
momentum, Hamiltonian, equation of motion as the Lorentz force law
in 3-vector and 4-vector notation
- Lecture 18 (Oct 19):
- Gauge invariance of [(e/c) A_i dx^i] term, equivalence to
[(1/c^2) A_i J^i]
- Gauge invariance and conservation of charge
- Gauge invariance forbidding [A_i A^i] term
- [F^ik F_ik] term in action: leading to Maxwell's equations with sources
- Possible [Ftilde^ik F_ik] term in action: disallowed by parity conservation
- Energy-momentum tensor: energy density, energy flow, stress tensor
as components
- Lecture 19 (Oct 21):
- Particle in a uniform electric field: catenary in relativistic motion
as opposed to parabola in non-relativistic
- Particle in a uniform magnetic field: difference between relativistic
and non-relativistic case
- Particle in combinations of electric and magnetic fields
- Particle in a coulomb potential: some features
- Tutorial (Oct 28)
- Lecture 20 (Nov 2):
- Lienard-Wiechert potential for a uniformly moving charge: pre-relativity
- Lienard-Wiechert potentials using covariant notation
- Electric and magnetic fields from LW potentials
- Solving the wave equation with Lorentz-transformed coordinates
to get the LW potentials
- No radiation for a uniformly moving charge
- Convection potential due to a moving charge,
force between two uniformly moving charges
- Lecture 21 (Nov 4):
- Cherenkov radiation: singularity in LW potential when [v < c/n]
- Calculation of power radiated by Cherenkov radiation
- Interpretation of Cherenkov radiation in terms of
constructive interference of wavefronts
- Lecture 22 (Nov 9):
- Radiation from an accelerated charge: treatment of partial derivatives
- Calculation of electric field, radiation component
- Calculation of magnetic field, radiation component
- Lecture 23 (Nov 11):
- Radiated power from a slowly moving accelarated charge,
angular distribution
- Radiation from a charge with acceleration and velocity in the
same direction, angular distribution of radiated power,
Bremsstrahlung
- Radiation from a charge in uniform circular motion,
synchrotron radiation as a energy loss mechanism and as a
source of high-frequency radiation pulses
- Assignment 3 (Given Sunday Nov 13,
Expected Monday Nov 28)
- Lecture 24 (Nov 18):
- Radiation reaction force for small velocities:
using Lierard-Wichert potential and its derivatives
- Validity of the radiation reaction force
- Radiation damping for ultra-relativistic particles:
analysis using 4-tensors
- Lecture 25 (Nov 23):
- Propagation of EM waves through matter: microscopic description
- Equation of motion for a bound electron excited by EM fields
- Scattering: Thomson, Rayleigh, resonant cross sections
- Absorption at the resonant frequency
- Complex refractive index and its frequency dependence
- Tutorial (Nov 25)