# Electrodynamics I (2020)

Classical electrodynamics is one of the crown jewels of human achievement. What Newton's laws did for the understanding of motion, Maxwell's equations did for a far more mysterious set of phenomena - it unified apparently disconnected phenomena related to electricity, magnetism, and light, and contributed to the discovery of special relativity. Electrodynamics is the simplest gauge field theory - a mathematical structure with beautiful and useful features that is now used to understand essentially all physical phenomena. An area that remains relevant to research owing to its myriad applications, it serves as a starting point for more fancy theory.

## Target Audience

This is the core course ED-1 or P-106 in the TIFR graduate school. If you already had a good course in electrodynamics before coming to TIFR, you should try to drop this course (email me before 10 Jan; Drop test is on 25 Jan) and directly take ED-2.

This page will be updated regularly with course-related information. Please check frequently.

I am available over email (PLEASE include the tag "ED2020" in the subject line to ensure my spam filter doesn't reject it). You can send **anonymous emails** if you like. Comments, criticism, cat-gifs, all are welcome.

## Administrivia

**Time:** 9:45 AM, Wednesdays and Fridays

**Venue: **AG 69

**First lecture:** 17 Jan 2020

**Credit policy:** 20% mid-term + 20% assignments + 30% term paper report and presentation + 30% from best of (end-term, mid-term, or term paper)

**Instructor:** Basudeb Dasgupta (A320)

**Tutor:** Sindhu (C338)

## Course Contents

1. Preliminaries and Maxwell's equations (4 lectures)

2. Electrostatics in vacuum and materials (8 lectures)

3. Magnetostatics in vacuum and materials (6 lectures)

4. Time-varying E and B fields, and their properties (4 lectures)

5. Waves (5 lectures)

Suggested term paper topics: Fundamentals of ED, Optics, Acceleration, Trapping, Radiative Transfer, Waveguides, Membranes, Super/Sub-luminal Light, etc.

## References

1. Modern Electrodynamics, Zangwill (Main Text - On Reserve in the Library; *Beware of typos!*)

2. Landau and Lifshitz Vol.2, Landau and Lifshitz

3. Landau and Lifshitz Vol.8, Landau, Lifshitz, and Pitaevskii

4. Classical Electrodynamics, Jackson

5. Feynman Lectures Vol.2, Feynman, Leighton, Sands

## Problem Sets

PS1: Mathematical Background and Maxwell Equations (assigned 1 Feb; due on 14 Feb)

PS2: Electrostatics (assigned 14 Feb; due on 28 Feb)

PS3: Magnetostatics (assigned 10 April; due on 1 May)

PS4: Time varying E and B fields and Waves (assigned 29 April; due on 25 May)

## Exams

Midterm: 29 March, 2-5 PM

Term Paper Presentation: 15, 20, 27, 29 May (in-class)

Endterm: 29 May (all day, at-home, optional)

## Lecture Summaries

Day Zero (17 Jan): Calibration and Course Overview

Lecture 1 (21 Jan): Mathematical preliminaries

Overview of electrodynamics

Vectors, tensors, gradient, divergence, curl, and Laplacian in Cartesian coordinates

Lecture 2 (24 Jan): Mathematical preliminaries

Vector identities and theorems

Drop Test: 25 Jan at 2 PM in A304

Lecture 3 (29 Jan): Intro to Classical ED

The 4+1 equations

Charge, charge density, current, and current density

When can we treat EM as classical

Lecture 4 (31 Jan): Intro to Classical ED

Is the photon massless? Is the force law really 1/r^2

Linear superposition

Lorentz averaging

Idealizations (what is a ground? boundaries)

Matching conditions

Maxwells equations and units

Problem Sheet 1 assigned (due on 14 Feb)

Lecture 5 (5 Feb): Setting up electrostatics

Electric field

Potential

Work

Potential Energy

Total Electrostatic Energy

Lecture 6 (7 Feb): Multipole expansion

Multipole expansion

Dipoles, potential, charge density of a point dipole, field, force and torque on dipole, energy

Quadrupoles

Dipole layer

Lecture 7 (12 Feb): More about multipoles

Traceless and spherical multipoles

Expansion of 1/|r-r'| in spherical harmonics or Legendre polynomials

Lecture 8 (14 Feb): Response of materials to electric fields

Conductors

Dielectrics

Lecture 9 (19 Feb): Solving Boundary Value Problems

Laplace and Poisson equations

Fundamental solution via Green's function: Basic Idea

Lecture 10 (21 Feb): Solving Boundary Value Problems

Images

Field near a conical point, corner etc.

Lecture 11 (26 Feb): Green's functions

Constructing the Green's function between two concentric spheres

Variations of the above approach, and relation to 1/|r-r'| expansion when region is free space

Note the relation between the Green's function and the solution via image method.

*No class on 28 Feb*

Midterm: 29 Feb at 2 PM to 5 PM

The test is closed book/notes

The test will cover material taught until end of Lecture 11

*No class on 4 and 6 March due to TIFR Graduate Admission Interviews*

Lecture 12 (11 Mar): Review

Review

Lecture 13 (13 Mar): Steady currents

Steady Currents and Summation Problems in Magnetostatics

**CORONAVIRUS COVID-19 Update (16 March): We will halt classroom lectures and move classes online. Check email for details.**

Lecture 14 (25 March, on zoom): Potential problems in Magnetostatics

Magnetic potentials

Solving for potentials

Multipoles

Lecture 15 (27 March, on zoom): Understanding Magnetostatic fields

Magnetic Fields and their peculiar topologies

Quadrupoles

Uses of B fields in lensing / focussing

"Paradoxes" about work done on moving current loops by inhomogeneous B fields

Lecture 16 (3 April, on zoom): Material response to static B fields

Magnetic materials have a B_self in response to B_ext

B_self is related to J_self = J_spin + J_orbital

J_spin is specified as curl of M_spin ~ curl of sum over point dipoles

J_orb is specified using probability currents; and its M_orb is not uniquely defined

The I=0, i.e., no current condition, and its implementation using J and K

Definition of potential and field A_M and B_M as sum over dipoles

Lecture 17 (8 April, on zoom): Simple linear magnetostatics and energy-momentum conservation

B and H fields

Constitutive relations

Force, Energy, Work

Using Maxwell's equations for magnetostatics in matter

Lecture 18 (10 April, on zoom): Review

Revision

Discussion of Term Paper Topics

Lecture 19 (15 April, on zoom): Time dependent E and B fields

Time dependent E and B fields

Notion of J_pol and M_pol

Flux Theorem

Displacement current, Induction, Lenz law, ..

Lecture 20 (17 April, on zoom): Linearity and Hierarchy of Scales in electromagnetism

Dispersion relations for linear PDEs

Quasistatic solutions of Maxwell's eqns.

Quasistatics in matter: Charge relaxation, Skin depth, Eddy currents

Lecture 21 (22 April, on zoom): Potential formulation of EM

Symmetries of electromagnetism

Gauge invariance

Potentials and how to choose a gauge

Equations for potentials

Lecture 22 (24 April, on zoom): Energy-Momentum of EM fields

Energy density, energy flux, Poynting theorem, Poynting vector

Linear momentum of EM fields, dyadic T, local conservation laws

Angular momentum of EM fields, dyadic M, local conservation laws

Feynman's paradox (FLP Vol.II Sec.17.4)

Hidden mechanical momentum of EM fields (if using NR mechanics)

Lecture 23 (29 April, on zoom): Waves

Wave equation

The scalar potential route to wave-like solutions

EM waves in vacuum (General, Plane, Transverse, Beam-like, Spherical)

Lecture 24 (1 May, on zoom): Waves

EM Waves in simple media

Lecture 25 (6 May, on zoom): Waves

Potential formulation

Transversality

Polarizations

Complex vectors

Spin and Orbital Angular Momentum of Transverse EM waves

Spherical waves and OAM

Lecture 26 (8 May, on zoom): Waves

Coherence

TE and TM waves

Lecture 27 (13 May, on zoom): Waves (Dispersion)

Linear response and time delay as the cause of dispersion

Lorentz model of dispersion

Term Paper Presentations (15 May)

Koshvendra (Ponderomotive force due to sunlight on a satellite)

Ritik (Magnetic lens)

Pruthvi (Birefringence)

Term Paper Presentations (20 May)

Rakeeb (Neutron radiation)

Avijit (Magnetic mirror and charge in E,B fields; see proton trajectory in Earth's B field below)

Himadri (BVP with realistic shapes; see a modelling of the Burj Khalifa below)

**Amphan Update (22 May): ** No class today (network problem for many)

**Term Paper Presentations (27 May)**

Himanshu (Synchrotron radiation; see a snapshot from a simulation of a moving charge below)

Rounak (EM tethered satellite; see thrust and drag of satellite below)

Ranjan (Dispersion)

**Term Paper and End of Course (29 May)**

Krishnendu (Physics of MRI; see simulation of T1 relaxation below)

Please submit all assignments to Sindhu

End of course feedback.

**Evaluation**

Congratulations, everyone! Let there be light!